Spring gift card give-a-way winner!

Congratulations to B.K.!! She is a 4th grade teacher and winner of my latest $25 gift card give-a-way. Thanks to my husband for getting the privilege of drawing the winner’s name. That’s his famous hand.

I will be contacting you, B.K., so start thinking about what type of e gift-card you would like to receive (Amazon, TPT, Staples, Wal-mart, etc.).

Thanks to the current subscribers, new subscribers, and commenters. I appreciate you and all you do for the children you teach. You are awesome!! Watch for another drawing in the fall.

I will continue with a few more posts, then I’ll take a break this summer. I hope you will use the time to check out my blog to find strategies that might be of help to you throughout the year. If you have a request for a topic, please let me know.

Cindy aka OK Math and Reading Lady

Multiplication Concepts Part 2: Arrays

by C. Elkins, OK Math and Reading Lady

Last week I posted my thoughts about multiplication strategies using the repeated addition strategy. This time I will focus on using arrays. Do you have some arrays in your classroom? Look for them with bookshelves, cubbies, windows, rows of desks, floor or ceiling tiles, bricks, pocket charts, etc. Students need to know arrays are everywhere! It is also very helpful for students to build arrays with objects as well as draw them. This assists students with moving from concrete to pictorial representations — then the abstract (numbers only) can be conceptualized and visualized more easily. Some good materials for arrays:

  • cubes
  • tiles
  • circular disks
  • flat stones
  • pinto beans (dry)
  • grid or graph paper
  • bingo stamper (to stamp arrays inside grids)
  • mini stickers
  • candy (Skittles, M&Ms, jellybeans)

Array Basics:

  1. Arrays form rectangular shapes.
  2. Arrays are arranged in horizontal rows and vertical columns.  This vocabulary is very important!
  3. The number of objects in each row (and column) in an array are equal.
  4. Arrays can be formed by objects, pictures, or numbers.
  5. Arrays can be described using numbers:  If there are 4 rows and 3 columns, it is a 4 by 3 array.
  6. The number of rows and number in each row are the factors. The product is the total.
  7. When an array is rotated, this shows the commutative property.

Ways to incorporate arrays into story problems:

  • Desks in a class (5 rows, 4 desks in each row)
  • Chairs in a classroom or auditorium (10 rows of chairs, 8 chairs in each row)
  • Plants in a garden (6 rows of corn, 8 corn plants in each row)
  • Boxes in a warehouse (7 stacks, 5 boxes in each stack)
  • Pancakes (3 stacks, 5 pancakes in each stack)
  • Cars in a parking lot (4 rows, 5 cars in each row)
  • Bottles of water in a crate (3 rows, 8 bottles in each row)
  • Donuts or cupcakes in a box (how many rows? how many in each row)

Activities to encourage concrete and pictorial construction of arrays:

  • Start off using manilla grid paper you probably have available with the construction paper supply at your school. This will help students keep their rows and columns even. Pose a problem and allow students to use manipulatives you have available to construct the array.  If you say, “Build an array for this multiplication problem: 3 x 5,” do they know the 3 refers to # of rows and the 5 refers to the number in each row?
  • Turn the paper after building the above array to see the commutative property. Now the picture shows 5 x 3 (5 rows with 3 in each row). The product is still 15.
  • Use the manilla grid paper along with bingo dobbers to create the array.  The grids can also be completed with mini stickers (I get them all the time in junk mail) or drawings.
  • When using pictures of arrays, direct your students to always label 2 sides of the array (the rows and columns). Try to label different sides of the array so it’s not always presented in the same format.
  • Find the product:  The whole point of using an array as a multiplication strategy is to visualize the rows and columns to help calculate the product. If students create rows and columns and then just count the objects one-by-one, then this does not accomplish the objective.  Show students how to skip count using the # of objects in the rows or columns. Believe me, students don’t always know to do this without a hint from the teacher.  Or better yet, before actually telling them to do this, ask students this question: “How did you get the total number of objects?” When you pose this question, you are honoring their strategy while secretly performing an informal assessment. Then when the student who skip counted to find the total shares their strategy, you give them the credit:  “That is an efficient and fast way to count the objects, thank you for sharing! I’d be interested to see if more of you would try that with the next problem.” Plus now students have 2 strategies.
  • Use the distributive property to find the product: Let’s suppose the array was 6 x 7.  Maybe your students are trying to count by 6’s or 7’s to be more efficient – but the problem is that counting by 6’s or 7’s is difficult for most students. Break up (decompose) the array into smaller sections in which the student can use their multiplication skills.  Decomposing into rows or columns of 2’s and 5’s would be a good place to start. This is the distributive property in action – and now the students have 3 strategies for using an array!! This is a great way to use known facts to help with those being learned.Here is a link to Math Coach’s Corner (image credited above) and a great array resource: Multiplication arrays activities from TPT $5.50.  Here is my FREE guided teaching activity to help students decompose an array into 2 smaller rectangles. Click HERE for the free blank template.
  • Use the online geoboard I described a few posts back to create arrays using geobands. Click here for the link: Online geoboard  Click here for the previous post: Geometry websites (blog post)
  • Try these freebies:  Free array activities from k-5mathteachingresources.com. Here’s a sample.

     

  • Play this game I call “Block-It.” This is a competitive partner game in which students must create arrays on grid paper. Click here for a FREE copy of the directions: Block-It Game Directions
  • Relate use of arrays when learning strategies for division and area.

In a future post I will show some ways to use manipulatives and pictures arrays for double digit multiplication problems. Stay tuned!!

Multiplication Concepts Part 1: Repeated Addition

by C. Elkins, OK Math and Reading Lady

The next few posts (until I take a break over the summer) will focus on the basic multiplication concepts one at a time. This will allow the opportunity to dig deeper into the concepts we want students to understand. This one will focus on the concept that multiplication is repeated addition. These posts will be helpful to teachers introducing multiplication to students in 2nd and 3rd grade as well as those in 4th, 5th, 6th and beyond who have missed some of these basic concepts. Future posts will focus on the area (array), set (equal groups), counting, and decomposing models as well as the associative and distributive properties.

Do your students know what the “times” sign means? They may hear it frequently, but not realize what it means. I like to interpret it as “groups of.”  So a problem like 3 x 4 can be said as “3 groups of 4.”

To show repeated addition, that same problem would be 4 + 4 + 4 = 12.

Repeated addition can be shown with numbers, and also with arrays and equal groups. These pictorial models are great for developing multiplication concepts (and will be topics of future posts). However, when students are presented with these models they often count the individual pieces one at a time rather than adding the same amount repeatedly. Observe your students to see how they are counting.

Do your students apply the commutative property of multiplication? This means if the problem is 3 x 4, it can also be solved by thinking of 4 x 3 (which is 4 groups of 3 OR  3 + 3 + 3 + 3). I want students to know even though the answers are the same, the way the factors are grouped is different. When used in a story, 3 x 4 is a different scenario than 4 x 3.

Do your students practice repeated addition, by combining 2 or more numbers? See the following for an illustration of 15 x 6:

Do your students apply the concept of repeated addition to multiple digit multiplication problems as well? I have witnessed students numerous times who only try a problem one way and struggle. For example, on a timed test I witnessed a 5th grader attempt the problem 12 x 3. I observed him counting by 3’s.  He was trying to keep track of this by skip counting by 3’s twelve times. I could tell he had to start over frequently, thus spending a lot of time on this one problem. It became obvious he had no other strategy to try. He finally left it blank and went on. Just think if he had thought of 12 + 12 + 12. This should have been relatively easy for a 5th grader.  He also could have decomposed it to this: (3 x 2) + (3 x 10).

Do your students always go to the standard algorithm when they could perhaps mentally solve the problem by repeated addition? If the problem was 50 x 3, are they thinking 50 + 50 + 50? Or are they using paper-pencil and following the steps?

What about a problem such as 45 x 4?  Using repeated addition, is your student thinking of 40 + 40 + 40 + 40 combined with 5 + 5 + 5 + 5? This is then solved as 160 + 20 = 180.

Students who are able to use repeated addition skillfully are showing a healthy understanding of place value and multiplication. This strategy also enhances mental math capabilities. Conducting daily number talks are highly advised as a way to discuss multiple ways to solve a given problem such as those mentioned above. Check out “Number Talks” in my category list for more information on this. Also check out some recommended videos about conducting number talks (above black bar “Instructional Resources”).

Remember the membership and comment challenge. $25 gift card drawing will be May 18th!!

Geometry Websites

by C. Elkins, OK Math and Reading Lady

There are several great math websites which might help you and your students with geometry and measurement standards such as area, perimeter, volume, surface area, angles, etc.  The ones I am recommending are interactive and often customizable.  Check them out!! (Each title can be clicked to take you directly to the linked website.)

  1. Geoboard by The Math Learning Center:  I love the concept of geoboards to help children create polygons and measure area and perimeter.  However, most teachers have ditched their physical geoboards. They are often in boxes relegated to the basement storage areas.  I get it, though.  They take up a lot of shelf space in the class, there aren’t enough rubber bands to go around (aka geobands), the kids misuse them or break them, they don’t stretch far enough, the pegs get broken, etc.

I think you will LOVE this app. Check out the little “i” on how to get the most use out of it, but it has 2 variations for the board size and you can show it with/without gridlines or numbers. There are different colored bands which you drag to the board and stretch to whichever pegs you need. You can shade in areas, copy, and rotate (which is helpful to see if 2 similar shapes are equivalent). There is also a drawing palette in case you want to freehand something or draw lines (and with different colors as well).

What are the possibilities with this?

  • Use with primary students to create squares, rectangles, and other polygons. The teacher can elicit different responses with directions such as:  Make a square. Make a different size square. Make a trapezoid. Are any of our trapezoids the same?
  • Creations can sometimes be recorded on dot paper – although I would reserve this for less-complicated shapes.
  • Count the pegs around the shape to determine perimeter. The teacher might ask students to create a rectangle with a perimeter of 10 (or 12, or another amount). How many different ways are there? Be cautious with diagonal connections because they are not equivalent to vertical or horizontal connections. Think of how you can get students to discover this without just telling them.
  • Show the gridlines to help students determine area.  Initially,  students may just count the squares inside the shape. Guide students to more efficient ways to figure this (multiplying, decomposing into smaller sections, etc.).
  • This app is also great for creating irregular shapes in which students may decompose into smaller rectangles or triangles. Then check them with the standard formulas.

2. “Cubes” at NCTM’s site (Illuminations):  This one is perfect for volume and surface area.

  • Volume:  You can use the gear symbol to select the size (l, w, and h) of the rectangular prism, or use the default ones shown. Then there are 3 tools used to fill the rectangular prism:  individual cubes, rows of cubes, or layers of cubes. I prefer using the layer tool to support the formula for volume as:  area of the base x height.  The base is the bottom layer (which can be determined by looking at the length x the width). The height is the number of layers needed to fill the prism. Once you compute the volume, enter it and check to see if it is correct.
  • Surface Area of Rectangular Prism:  To calculate the surface area, you must find the the area of each face of the prism. Again, you can customize the size using the gear tool.  I prefer this as the shapes shown randomly often are too small to see. Yes, there is a formula for surface area — but conceptually we want students to note the surface area can be thought of in three parts. With a click on each face, this app opens (or closes) a rectangular prism into the 6-faced net making it easier to see the equal sized sections:
    • Area of the front and area of the back are the same
    • Area of the top and area of the bottom are the same
    • Area of each side is the same
    • Be sure to explore what happens when the prism is a cube.

3.Surface area with Desmos:  This link provides an interactive experience with surface area, using a net. This time, the three visible faces of the prism are color coded, which helps with identifying top / bottom; front / back; and side / side. The prisms on this site are also able to be changed so students can see how altering one dimension affects the surface area.

4. “Lines” on GeoGebra

5. “Angles” on GeoGebra

6. “Plane Figures” on GeoGebra

These three may be more relevant to middle school math standards.  Check them out!!  Also take a look at the “Resources” link (left side of web page).  There are plenty of other good links for arithmetic standards as well – too many to list here.  You may have to create a log-in, but it’s FREE!

Enjoy!  Do you have other websites to recommend? Let us know.

Graphic Organizers for Math

by C. Elkins, OK Math and Reading Lady

Here are some cool graphic organizers for your math files!  Make sets of them, laminate or put in plastic sleeves, and use them over and over again!  Graphic organizers help students stay organized and teach them how to complete problems neatly. They are also a great way for students to show different strategies for the same problem. While primary students may need an already-made graphic organizer, intermediate students should be taught how to duplicate them on their own to use whenever the need arises – so the simpler, the better! With repeated use, students are more likely to utilize them regularly in their daily work (and on their scratch paper with assessments).

This one has ten frames and part-part-whole models. In my opinion, these are essential when working with K-2 students because they help children with subitizing, number bonds, and addition / subtraction facts.  If you are using Saxon, you are missing these important strategies!!:

Here’s one to show fractions (area, set, length models)

Need a template for students to make arrays? This one is ready!  I love showing students how to break an array into smaller parts to see how multiplication (or division) facts can be decomposed.  Example:  Make a 6 x 7 array.  Section off a 6 x 5 part. Then you have a 6 x 2 part left over.  This proves:  6 x 7 = (6 x 5) + (6 x 2).  Or — 6 x 7 = 30 + 12 = 42

This graphic organizer shows 5 different multiplication strategies using 2 digit numbers, and a blank one for students to record their thinking. Very handy!!  One of my favorite strategies is partial products. I highly recommend this one before going to the std. algorithm because students decompose the problem by place value and must think about the whole number and not just the parts.

Do your students need something to help them see the different models for a decimal? Try out this graphic organizer. Students will utilize the pictorial forms as well as the abstract.

Do your students know that .7 (or 7/10) is the same as .70 (or 70/100)?  Using this dual set of tenths and hundredths grids will help them see why this is true!

Be sure to check out my FREE templates and organizers (see black bar above “links . . .”)  Please share your favorite graphic organizers for math!  Enjoy!!

Graphic Organizers for Literacy

by C. Elkins, OK Math and Reading Lady

I highly recommend the use of graphic organizers. The purpose is to help students organize information with regard to different text structures:  

  • Compare and contrast
  • Cause and effect
  • Details / Descriptive
  • Problem and solution
  • Sequence

Graphic organizers are also helpful with standards such as:

  • Main idea
  • Summarizing
  • Character analysis
  • Story elements

Graphic organizers help organize the student’s thinking, and assist with note-taking. The visual pictures created help the student “see” the text structure, recall details, state the main idea, and summarize the selection.

Here are links to some sites I think provide good quality graphic organizers which can be utilized with a variety of situations:

  1. This one is more primary oriented: https://www.eduplace.com/graphicorganizer/
  2. This one is oriented more for 3rd and above: http://www.educationoasis.com/printables/graphic-organizers/
  3. This one is a FREE resource at TPT (as pictured above) that supports each of the 5 text structures: https://www.teacherspayteachers.com/Product/Non-Fiction-Text-Structures-Flip-Flap-and-Graphic-Organizers-Freebie-1777102

I have also linked these in “Instructional Resources” and in the categories list on my blog.  Enjoy!!

Division Basics Part 3: Repeated Subtraction and # Line

by C. Elkins, OK Math and Reading Lady÷

In my opinion, the process of repeated subtraction is very important for students to practice. With repeated subtraction, we are actually asking this question:  “How many _____ in _______?”  If the problem was 20÷4, we can ask, “How many 4’s are in 20?”  The process is to keep subtracting 4 (using concrete, pictorial, and abstract methods) until zero is reached.  This would be done 5 times — thus, 20 ÷ 4 = 5.

Much like multiplication, there are different aspects of division children should become familiar with.

  • Arrays 
  • Equal Groups
  • Repeated Subtraction
  • Number lines
  • Skip counting

The focus today will be to help children understand how repeated subtraction can assist with the division process (using manipulatives, drawings, and paper-pencil methods). The template pictured here is FREE from: Multip. and Division templates FREE from Number Two Pencils @ TpT

The reason the repeated subtraction strategy is important is because this is what we are really asking students to do when they encounter long division or partial quotient problems. With the problem 100 ÷ 4, the question is, “How many 4’s are in 100?” If the repeated subtraction process is used, the answer is of course, 25.  But subtracting 4 twenty-five times is not very efficient.  So we want the student to get closer to 100 and subtract larger amounts than 4 at a time. The partial quotients method would allow the student to do this in chunks.  1 solution could be to subtract 40 (ten 4’s), subtract another 40 (ten more 4’s), subtract 20 (five 4’s).  See picture below: Continue reading

Division Basics Part 2: Equal Groups

by C. Elkins, OK Math and Reading Lady

Last post featured division using arrays and the area model.  This post will focus on helping children see division as equal groups. Most of us have used the “plates of cookies” analogy to help kids see how to represent equal groups in a drawing.  I will just take that a few more steps to increase efficiency.

Much like multiplication, there are different aspects of division children should get familiar with:

  • Arrays 
  • Equal Groups
  • Repeated Subtraction
  • Number lines
  • Skip counting

In this post, I will break down the benefits of equal groups models to help children understand division (and how it is related to multiplication). Check out the freebies within this post.

If you haven’t utilized this book with your students, please try to find a copy!  It’s called The Doorbell Rang by Pat Hutchins.  In this story, Ma makes some cookies to be split between the kids.  Then the doorbell rings and more kids come, so the problem has to be refigured. This scenario repeats. As a class, you can duplicate the story with a different # of cookies and children.

Another great story emphasizing equal groups (as well as arrays) is the story One Hundred Hungry Ants by Elinor Pinczes.  In this story, 100 ants are on their way to raid a picnic. They start off in one straight line (1 x 100), but then rearrange into different equal groups to shorten the line (2 lines of 50, 4 lines of 25, etc.). A nice project after reading this book is to see how many ways a different given # of ants (or other animals / objects) can be divided into equal groups / rows.

 

By clicking on the links for each book above, you will be taken to Amazon for more details.

As I mentioned earlier, many children’s view of equal groups regarding division is to use manipulatives and/or draw circles / plates to match the divisor and then divide up the “cookies” equally in these groups.  Let’s say you had this problem: “There are 12 cookies to be divided onto 3 plates equally.  How many cookies would go on each plate?” As you observe the students:

  • How are they dividing up the cookies? One at a time, two at a time, randomly, trial and error?
  • Are the “cookies” scattered randomly on the plate / circle?  Or, are they arranged in an easy-to-see pattern so they are easily counted (by the student and yourself as you walk around the room)?
  • Are the students able to verbally tell you how they divided them?
  • Are the students making the connection to multiplication by noting that 3 x 4 = 12?
  • Can they solve similar problems using language other than plates / cookies?
    • Try shelves / books; trays / brownies; buildings / windows; flowers / petals; students / rows of desks, stars / # of points; aquariums / fish; boxes / donuts; etc.

Use of manipulatives of various types (cubes, tiles, counters) is important for children to have their hands on the objects being divided. This is how they work out their thinking. Then work toward paper/pencil drawings before going to the abstract use of numbers only.  This doesn’t have to be done in separate lessons, however. There is great value for children to see how the concrete, pictorial, and abstract representations all work together.

Also, help children list synonyms for the dividing process:  distribute, share, split, separate, halve, quarter, partition

Here are a few strategies I believe help make the equal groups process more efficient: Continue reading

Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady

Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics.  Students as young as first grade actually start thinking about division when working on fraction standards such as:  Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

  • Arrays 
  • Equal Groups
  • Repeated Subtraction
  • Number lines
  • Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun FREE game (Block-It) at the end of the post.

What is an array?  An array is a rectangular model made up of rows and columns.  When an array is constructed, the factors are represented by the number of rows and columns.  So, do your students know the difference in a row and column?  (Rows go horizontally, while columns are vertical.)  These are important math terms students should be using.

  • Give students experience constructing arrays with manipulative objects (tiles, chips, cubes, etc.):
    • You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows.  How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the dividend (the total amount you started with). The # of rows is the divisor (how it was divided).  The quotient is the result (in this case the # of columns).
    • You can also be a little more open ended such as:  “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc.
  • Give students experience drawing arrays:
    • You can be specific or open-ended (as above).
    • Children can free-hand draw or use grid paper.  If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.”
  • Give students experience using pre-drawn arrays:
    • Students should label the sides of the array with numbers.
    • Use the numbers shown to determine the fact family.  Example:  3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3
  • After the array is made, ask questions or explore more such as:
    • How many 3’s are in 12? (count the columns)
    • How many 4’s are in 12? (count the rows)
    • Circle the rows and / or columns to see the groups more easily.
    • Help children make up story problems to match the array:  “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

Continue reading

Text Structures Part 4: Sentence Frame Posters

by C. Elkins, OK Math and Reading Lady

Today’s post is the result of a project I have been working on for awhile.  I created some posters you can use in your classroom which feature sentence frames connecting text structure to the skills of main idea and summarizing.

Here are samples of 2 of the Main Idea posters. Get the full set here FREE: Text Structure Main Idea Posters CE-2019  There are 10 posters (1 Main Idea and 1 Summarizing poster for each of the five text structures).  If you have suggestions for improvement, please let me know.  I want to make these usable for YOU!

Text Structures Part 3: Sequence and Descriptive

by C. Elkins, OK Math and Reading Lady

Welcome back to the third text structure post.  Today’s focus will be on sequence / chronological order and descriptive text structures. Here are some graphic organizers to keep in mind.

Sequence / Chronological Order

1. Sequence refers to a particular order in time. This can be:

  • Information presented minute by minute, hourly, weekly, monthly, yearly, etc.
  • Providing information by dates (a timeline)
  • Steps of how to complete something (first, second, third, etc.)
  • A retelling of events in the order they happened: First, next, then, finally or beginning / middle / end.  It may be helpful to use a “retelling rope”.   Use a section of rope or nylon cord (approx. 1 foot long). Tie several knots along the length of it (3-5). At each knot, retell part of the story or events in sequence.
  • Observing how things / people have changed over time
  • Non-fiction and fiction selections
  • Arranging events in order using pictures

2. Connecting sequence to strategies:

  • Predict what will happen next in the sequence.
  • Visualize the steps involved.
  • Make personal connections regarding your own experience with the sequenced topic.

3. Sequence / Chronological order main idea / summarizing sentence frames:  Suppose I read an article telling about the seasonal journey of a pod of whales.  Again, the topic is whales — but this is NOT the main idea.

  • (Main idea):  Whales travel to different locations each season to find food and a mate.
  • How to ________ step by step.
  • The timeline of _________________.
  • There are several steps to ______________. First, _________. Then, ___________. Last, ________.
  • The life cycle of __________.
  • Many things happened during _____________’s life.
  • (Summarize): Whales travel to different locations each season to find food and a mate. In the spring, they ________. In the summer, ______________.  In the fall, _____________. In the winter, _________.
  • To make ________, follow these steps: ________________.
  • The life cycle of a ___________ includes these stages: _______________.
  • Many things happened during _____________’s life. In (year), he/she_____________. After that, _____________. Then, ________________. Finally, ___________________.

Descriptive Text Structure

1. Descriptive structures give details.  These can be:

  • Details or descriptions about a person, a place, a thing, an idea, an animal, an event, etc.
  • A web graphic organizer is a good model to visualize, with the topic in the center and the supporting details branching outwards.

2. Connecting to strategies:

  • Visualize what is being described, especially if there are no pictures or photos in the text.
  • Ask questions about the topic such as:  “I wonder . . .”
  • Analyze the point of view:  What is the author’s point of view. Is he/she presenting a one-sided view of the details presented?
  • Make connections to the topic.

Continue reading

Text Structures Part 2: Cause and Effect + Problem / Solution

by C. Elkins, OK Math and Reading Lady

Welcome back to part 2 regarding Text Structure.  As I mentioned before, pairing a text with a graphic organizer to help highlight the structure can be very helpful to frame the main idea and summary. When a graphic organizer is used often, then students begin to visualize them and organize their thoughts mentally as well.  And still better . . . combining text structure instruction with reading strategies such as visualizing, questioning, making connections, and predicting will lead to higher comprehension.

Today’s focus will be on two other text structures:  Cause / Effect and Problem / Solution.  These two are related, but often confusing to students. Look for some resources at the end of this post.

Cause and Effect:

Cause:  The reason why something happened.

Effect:  The result — what happened?

A cause / effect text structure can show 1 cause and several effects.  Example: An earthquake can be the cause of many events (damaged structures, ruptured pipes, injuries, accidents, tsunami, etc.).  When this is the case, it may be simpler to identify the cause first, then identify all of the effects.

On the other hand, a cause / effect text structure can show several causes for 1 effect.  Example: Some animals are endangered (effect) due to these causes: pollution, loss of home environment due to destruction of their habitat, weather, disease. When this is the case, it may be simpler to identify the effect first, then identify all of the causes.

Other notes about teaching cause / effect:

  • This text structure can apply to non-fiction as well as fiction texts.
  • Because many cause / effect relationships require defining the problem (which could be the cause and sometimes the effect as well), students often get confused and identify the structure as problem / solution.
  • Not all cause / effect relationships are about problems. Example:  I love my grandson’s drawings (cause), so I hang them on the refrigerator (the result / effect). No problem here!
  • While most anchor charts posted online provide key words for the cause / effect structure (because, reason, since, as a result, etc.), I would suggest limited use of them especially when first analyzing the structure. I have found when mentioning them first, students often just start looking for those key words and are not truly reading the text.  And . . . those words can also be found in almost any text anyway.  You don’t want kids to reduce this to a competition: “How many time did I find the word because?” Those words don’t even have to be there for there to be a cause / effect relationship.
  • Use a graphic organizer with an arrow connecting the cause to the effect.
  • Even young students can understand simple cause / effect relationships presented in stories.  Discuss the causes and effects and/or write them as a shared writing experience. See some resources below for great books on this structure.

Combining with strategy work:

  • Visualize actions of the subjects in the text to picture the causes and results.
  • Make connections to things, places, events in the text you have experienced. Make predictions based on those experiences regarding why things happened.
  • Help students ask questions about the text.  They should be wondering why certain things happen, or what caused what. Learn to read on (or check other resources) to see if those questions get answered.
  • Make inferences about the causes in the text. Read between the lines.

Connecting to main idea and summary. Supply some sentence frames so students are using compare/contrast language. Suppose an article describes the causes of beached whales. The topic is whales — but that’s NOT the main idea:

  • (Main Idea): There are many reasons a whale becomes beached.
  • (Summary):  There are many reasons a whale becomes beached such as low tide, changes in ocean currents, chemicals in ocean water, and disorientation due to man-made sonar devices.
  • (Main Idea):  There are many causes for _________________________.
  • (Main Idea): The main cause for ____________ is _______________.
  • (Main Idea): There are several reasons why __________ decided to ___________.
  • (Summary):  There are many causes for __________________ such as _________________.
  • (Summary):  When _______________ happens, the result(s) are ___________________.

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Text Structures Part 1: Compare and Contrast

by C. Elkins, OK Math and Reading Lady

I have come to realize just how important knowledge of text structures is to almost all of the other comprehension skills and strategies. So that will be my focus for the next few posts — how this text structure connection relates to main idea, summarizing, note-taking, and writing. This post will feature the compare and contrast text structure (and some resources at the end of this post).

What are the text structures? Most sources consider the following 5: (Picture from Mrs. M’s Style. Here’s the link on Pinterest:  Text Structure Mini Anchor Chart)

  1. Compare and Contrast
  2. Cause / Effect
  3. Sequence
  4. Details / Description
  5. Problem / Solution

When I see reading texts that indicate the week’s skill is text structure, I cringe a little bit.  Why? Well, if you are teaching all 5 of them – that’s too much to digest in one week.  Here’s what I think is much more practical:  Teaching about text structures should occur with each and every reading selection — and refer to the structure that is most evident regarding that selection.

Here’s an example of what the teacher might say:  “This week we are reading an article titled Whales and Dolphins.  This article will compare and contrast whales with dolphins. Compare and contrast is a text structure in which the author will tell ways the whales and dolphins are alike and different from each other.”

How can I further connect this to comprehension and text structure?

  • Venn Diagrams or T-charts are helpful graphic organizers regarding compare/contrast text structure. Student can take notes using the graphic organizer. The idea is that with frequent use, students can eventually visualize this graphic organizer model in their head. Then this visual model serves as a thought organizer when they are not able to physically utilize one.
  • I can direct my questions to focus on this text structure such as: “On page 37, can you find one way the author compared whales to dolphins?”  “On page 39, the author told 3 ways the whales and dolphins are different. What did he say?”

How can I further connect this to help students with the main idea and/or a summary of a compare/contrast article?  Using information from notes on the Venn Diagram, students can use sentence frames like these:

  • This article compared _____________ to ______________.  (main idea)
  • This article compared ___________ to _____________.  Whales and dolphins are alike because _____________ and they are different because ___________________. (summary)

Continue reading

Geometry Part 8: Area and Perimeter (cont’d)

by C. Elkins, OK Math and Reading Lady

This post features 3 more area and perimeter misconceptions students often have. I have included some strategies using concrete and pictorial models to reinforce the geometry and measurement standards. Refer to Geometry Part 7 for 2 other common misconceptions.

Also, check out some free resources at the end of this post!!

Misconception #3:  A student only sees 2 given numbers on a picture of a rectangle and doesn’t know whether to add them or multiply them.

  • Problem:  The student doesn’t know the properties of a rectangle that apply to this situation — that opposite sides are equal in measurement.
  • Problem:  The student doesn’t see how counting squares can help calculate the area as well as the perimeter.

Ideas:

  • Give the correct definition of a rectangleA quadrilateral (4 sides) with 4 right angles and opposite sides are equal.
  • Give the correct definition of a square:  A quadrilateral (4 sides) with 4 right angles and all sides are equal. From this, students should note that squares are considered a special kind of rectangle.  Yes, opposite sides are equal – but in this case all sides are equal.
  • Using square tiles and graph paper (concrete experience), prove that opposite sides of a rectangle and square are equal.
  • Move to the pictorial stage by making drawings of rectangles and squares. Give 2 dimensions (length and width) and have students tell the other 2 dimensions.  Ask, “How do you know?” You want them to be able to repeat “Opposite sides of a rectangle are equal.” With this information, students can now figure the area as well as the perimeter.
  • Move to the abstract stage by using story problems such as this:  Mr. Smith is making a garden. It will be 12 feet in length and have a width of 8 feet.  How much fence would he need to put around it? (perimeter) How much land will be used for the garden? (area).
  • Measure rectangular objects in the classroom with some square units.  Show how to use them to find the perimeter as well as the area using just 2 dimensions.  Ask, “Do I need to fill it all the way in to determine the answer?”  At the beginning – YES (so students can visualize the point you are trying to make). Later, they will learn WHY they only need to know 2 of the dimensions to figure the area or perimeter.

Continue reading

Geometry Part 7: Area and Perimeter

by C. Elkins, OK Math and Reading Lady

Today’s topic is the measurement of area and perimeter.  Even though these may be considered measurement standards, they are highly connected to geometry (such as the attributes of a rectangle). Check out previous posts in my Geometry series (Composing and Decomposing) for other mentions of area and perimeter.

Misconceptions provide a window into a child’s thinking.  If we know the misconceptions ahead of time, we can steer our teaching and directions to help students avoid them. I will go through several misconceptions and some strategies and/or lessons that might address them. Misconceptions #1-2 appear in this post. Misconceptions #3-5 will be featured in next week’s post.

Misconception #1:  A student hears this:  “We use area to measure inside a shape and perimeter to measure around a shape.”

  • Problem:  The student doesn’t know how to apply this definition to real situations which require the measurement of area and/or perimeter.
  • Problem:  The student may think, “Since perimeter measures the outside edge, then area means to measure the inside edge.”
  • Problem:  Students confuse the two terms.

Ideas:

  • Brainstorm with students (with your guidance) examples of the need for area and perimeter. Use these scenarios as you solve concrete or pictorial examples.
    • Area:  garden, room size, ceiling tiles, carpet, rug, floor tiles, football field, tv screen, wallpaper, wall paint, etc.
    • Perimeter:  picture frame, fencing, floor trim, wallpaper border, bulletin board border . . .
  • Show this diagram which emphasizes the concept that area measures the entire inside surface using squares of different sizes (cm, inch, foot, yard, mile), while perimeter measures the rim / edge / around an object or shape usually using a ruler, tape measure, or string.  
  • Try this project: Use graph paper and one inch tiles (color tiles) to concretely make shapes with a given area.  Example, “Build a rectangle with an area of 24 tiles.” Specify there can be no holes in the rectangle — it must be solid. Students can experiment while moving the tiles around. Then trace the rectangle and cut it out. With this practice, students are also focusing on arrays and multiplication.  Did they find 4 ways? (2 x 12, 3 x 8, 4 x 6, 1 x 24)? Did they find out a 3 x 8 rectangle is the same as an 8 x 3 rectangle (commutative property)?
    • Note:  Make sure students stay on the given lines when tracing their shapes and cutting. I found this to be difficult for many students even though I said explicitly to “Stay on the lines when you trace.”
  • NO – this is not a solid rectangle. No holes allowed.

  • Similar to the above:  Use the tiles and graph paper to create irregular shapes with a given area – meaning they don’t have to be rectangular. Give the directive that tiles must match at least one edge with another edge (no tip to tip accepted).  And, same as above — no holes in the shape. You can even assign different areas to each small group.  Compare shapes – put on a poster or bulletin board.
  • Using the same shapes made above, determine the perimeter.  I suggest placing tick mark to keep track of what was counted. With this lesson, students will hopefully realize shapes with the same area do not necessarily have the same perimeter. Advise caution when counting corners or insets – students usually miscount these places. 
  • Try this project:  Design a bedroom with dimensions of (or square feet of): ___________ Include a bed, and at least one other piece of furniture or feature (nightstand, dresser, desk, shelf, chair, closet, etc.)
    • The student can use smaller scale graph paper with 1 square representing 1 square foot.
    • Let them have time to “work it out” and practice perseverance as a rough draft before making their final copy.
    • Together, use the square foot construction paper pieces to determine an appropriate size for a bed and other furniture. Students must think of it from an overhead perspective. When I did this with a class, we settled on 3 ft. x 6 ft bed for a total of 18 square feet. We did this by laying the square foot papers on the floor to see in real life what this size looked like.
    • Label the Area and Perimeter of each item in the bedroom.
    • The items in the bedroom could also be made as separate cut-outs and arranged / rearranged on the “floor plan” to see all of the different ways the room could look.
    • Students in 3rd grade might want to use whole units, while 4th and up might be able to use half-units.
  • On a test, area answer choices would include ” ____ square inches” or “inches squared” or “inches²”  Answer choices for perimeter will omit the word “square.”

Continue reading

Geometry Part 6: Angles and Lines

by C. Elkins, OK Math and Reading Lady

When working with students on geometry lessons involving angles and lines, I notice many misconceptions. So . . . I thought I would share them with you on this post. Some activity ideas and freebies are located at the end of this post.

Right Angles:  

  1. Students can only see the 90° angle if it is presented in the direction as a capital L.
    • Try turning the angles in different positions.
    • It is still considered a “right” angle even though it is turned to the left, up, or down.
  2. Students are told if they can draw a square inside the angle, then it is a right angle. So if it looks “squarish” to them, they think it’s a right angle.
    • Show them how to put the square corner of a piece of paper or index card into the angle to check. Take time to have them practice – don’t assume they know how.

Acute Angles (angles less than 90°):

  1. Students often can’t tell if the angle is <90° if it is oriented upside down or if one of the rays is not aligned horizontally.
    • Show how to put the square corner of a piece of paper into the angle to check. If the paper covers up the angle, it is <90°.
  2. Students are told an acute angle “is a cute little angle.”  I am guilty of having done this in the past. But if a student sees any angle made up of short lines, they may interpret it as “little” or acute.
    • Remind them it’s the size of the angle that makes it acute, not the size of the lines.
  3. While right angles are exactly 90°, students may expect an acute angle to be given a specific number.
    • Acute angles range from 1° to 89°.

Obtuse Angles (angles greater than 90°): Continue reading

Geometry Part 5: Composing and Decomposing 3D Shapes (+ surface area)

by C. Elkins, OK Math and Reading Lady

Composing and decomposing 3D shapes should help your students become more familiar with their attributes. Here are a few activities to help.  With emphasis on hands-on methods, examining real 3D shapes may help students find edges, vertices, and faces better than pictorial models.

  1.  Nets of 3D shapes are the least expensive way to get a set of 3D objects in each child’s hand, especially since most classrooms just usually have 1 set of plastic or wooden 3D shapes.
  2.  Build cubes and rectangular prisms using blocks or connecting cubes.
  3. Construct / deconstruct prisms using toothpicks, straws, coffee stirrers, craft sticks, or pretzel sticks as the edges. For the vertices, use clay, playdough, gum drops or slightly dried out marshmallows.
  4. Lucky enough to have a set of tinker toys? Or Magna Tiles? (We got our grandson some Magna Tiles and he loves them!  These tiles have magnetic edges which can hook together in an instant. Creating a cube, rectangular prism, pyramid, etc. is easy!  They are kind of expensive, but very versatile and creative.)
  5. Teach students how to draw 3D shapes. When composing a 3D shape, a student becomes more aware of the 2D faces, the edges, and the vertices they are drawing. Plus, if needed the student can draw the 3D shape on paper to assist them if taking a computer based assessment.  Here is my tutorial (below), but I’ll also include a couple of good websites in case you are 3D challenged. Click HERE for the pdf of the templates below.
  6. Observe how students count the edges, vertices, and faces.  If they are randomly trying to count them, they likely will be incorrect.  When needed, show them how to be methodical with their counting (ie: When counting the edges of a cube, run your finger along the edge as you count. Count the top 4 edges, then the bottom 4 edges, then the 4 vertical edges = 12.)

One of my favorite lessons regarding decomposing shapes is when teaching students (5th grade and up) how to measure surface area.  Click HERE for the free pdf guide for creating the rectangular prisms shown below.  It includes a blank grid so you can create your own (all courtesy of http:illuminations.nctm.org using their “dynamic paper” lesson). Continue reading

Geometry Part 4: More Composing and Decomposing

by C. Elkins, OK Math and Reading Lady  

There are so many good ways to help students compose and decompose shapes (2D and 3D), so I will focus on some more by using tangrams and 2D paper shapes. In case you missed it, my last post focused on ways to use 1″ color tiles and pattern blocks to compose and decompose shapes. Click HERE to link back to that.

  1. Give students paper shapes of these polygons:  rectangle, square, hexagon, trapezoid, rhombus. Click here for a FREE pdf copy: Decompose and Compose Polygons.
    • Students should color each paper shape one solid color (a different color for each shape). My advice is to use light colors because they will be drawing lines on the shapes and light colors enable them to see the lines.
    • Model how to draw 1 or 2 lines to decompose the shape into smaller shapes.  For first and 2nd grade, I recommend you show them how to use at least one corner of the shape to connect to another corner or side using a straight edge or ruler. This way the newly created shapes will resemble ones they already know (triangle, trapezoid, rectangle). Older students can be given a little more leeway — their decomposing may result in other more irregular polygons. Here is one way to decompose.
    • Cut apart on the lines. Have students put their initials or name on the back of each piece (in case it gets separated or ends up on the floor).
    • Each student puts their cut-up pieces in a baggy for safe-keeping. Then the student can take them out and try to compose them back into their original shapes.  This is where the color-coding comes in handy (all the yellow go together, all the green, etc.).
    • Students can trade their baggies with others to compose their shapes.
    • When students are done with the shape puzzles, they can glue them back together on background construction paper (or take them home for practice, or keep at school for ongoing work).
    • Discuss together how many different ways these shapes were decomposed using 1 or 2 lines.
  2. Use the book, “The Greedy Triangle” by Marilyn Burns as a springboard to compose other polygons using various numbers of triangles.  In this book, the triangle keeps adding a shape to himself (after a visit to the “Shapeshifter”). There are many good pictures in this book illustrating common things with the named shape.  This is also a great way to connect art to math. You can start with squares which the students must cut in half on the diagonal, or start with pre-cut triangles. Length of edges must match. Level 0 students can just try out different combinations. Level 1-2 students would analyze the properties more and name the new shapes. You can even emphasize symmetry (as I have shown with the bottom row). Here is the link to the full article about this wonderful activity. Math Art: The Greedy Triangle Activity

Continue reading

Geometry Part 3: Composing and Decomposing

by C. Elkins, OK Math and Reading Lady

Composing and decomposing geometric shapes (2D and 3D) should be centered around concrete and pictorial methods. In this and upcoming posts, I will illustrate some methods using various manipulatives and line drawings which help students take a shape apart or put shapes together. If you refer back to  Geometry Part 1: The Basics, all grade levels KG-5th have standards dealing with this issue. Some of the experiences I plan to share will also help students relate to multiplication, division, fractions, area, and other geometry concepts (such as rotations, reflections, slides).

Refer to Geometry Part 2: van Hiele levels to determine if the activities you are choosing are appropriate for Level 0, 1, or 2 students.

One Inch Color Tiles:

1.  Can you make a larger square out of several individual squares?

  • Level 0 students will be using the visual aspect of making it look like a square.
  • Level 1 students will be checking properties to see if their squares are indeed squares (with the same number of tiles on each side).
  • Level 2 students will be noticing they are creating an array (ex: 3 x 3 = 9) and perhaps learning about squared numbers. 3 squared = 9. They might be able to predict the total number of tiles needed when given just the length of one side.

2.  How many rectangles can you make using 2 or more squares? (Level 0-1)

  • Level 1:  Are the green and blue rectangles the same size (using properties to determine)?

Continue reading

Geometry Part 2: Learning Continuum (van Hiele)

by C. Elkins, OK Math and Reading Lady

Today’s post will focus on an aspect of geometry involving levels of thought.  We know PreK or KG students aren’t ready for formal definitions regarding shapes. Starting about 2nd grade, students might be ready to describe shapes using specific attributes. Pierre and Dina van Hiele are Dutch math educators who have an excellent way to describe how children move through these geometric thinking levels.  They are called “The van Hiele Levels of Geometric Thought.” Click on this link for a full description:  The van Hiele Model   Also – some good resources at the end of this post.

I became interested in these levels as I was doing research about better ways to help students master standards in Geometry.  (See more information below regarding these levels.)  Am I supposed to teach them the “proper definition” of a square in KG? At what point should students begin to understand the specific properties of a square – that it is actually a specific type of rectangle. And . . . when is it appropriate to help students realize that a square is also classified as a rhombus, rectangle, parallelogram, and quadrilateral? Click on the link for a pdf copy of the chart below: van Hiele Levels 0, 1, 2

What can we as teachers do to help them move through the levels? According to van Hiele, the levels can’t be skipped – children must progress through each hierarchy of thought. So while there is no grade level attached to these levels,  I like to think of the Visualization Level as the beginning point most appropriate for PreK-1st or 2nd grade students. Level 1 thinking might surface in grade 2 or 3 (and up). Students capable of Level 2 thinking may start with 3rd – 8th grade. Students in high school geometry might function at Level 3 thinking.

One of the properties of the levels he described has the name of “Separation.” The article linked above states: A teacher who is reasoning at one level speaks a different “language” from a student at a lower level, preventing understanding. When a teacher speaks of a “square” she or he means a special type of rectangle. A student at Level 0 or 1 will not have the same understanding of this term. The student does not understand the teacher, and the teacher does not understand how the student is reasoning. The van Hieles believed this property was one of the main reasons for failure in geometry. Teachers believe they are expressing themselves clearly and logically, but their Level 3 or 4 reasoning is not understandable to students at lower levels. Ideally, the teacher and students need shared experiences behind their language.

Here’s a closer look at the levels. Continue reading