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:

With the std. long division algorithm, students also must think of division as repeated subtraction. You wouldn’t believe the number of “ah-ha’s” I get from students when I show them this concept!!  Here’s a link to a previous post on long division. Making sense of division

So, how do we encourage this strategy with new learners using basic division facts? Check these out:

  1. Manipulatives:  If the problem is 20 ÷ 4, start with 20 objects.  Then take away 4 at a time by moving them to the side and forming a group. Repeat until all 20 objects have been taken from the original group. How many groups were made? This is a quick, meaningful any very efficient method to help students actually see equal groups being constructed. A must before going to drawings or paper pencil methods.
  2. Drawings:  If the problem is 15 ÷ 3, draw 20 objects.  Cross off 3 at a time and keep track (ex: tallies). Keep crossing off 3 at a time until all 15 objects have been crossed off. How many times was this done?
  3. Number line:  If the problem is 10 ÷ 2, draw an open # line and label with 10 points (0-10). Starting with 10, jump backward 2 at a time until 0 is reached. Count how many jumps were made. This is more effective with dividends 20 or less due to space and amount of time this takes. Consider a vertical # line as well.
  4. Subtraction:  If the problem is 56 ÷ 8, write the # 56, then subtract 8 repeatedly until there is zero left. Keep a running total. Common errors with this method are failure to follow regrouping steps and poor calculations.

I came across this FREE resource from Number Two Pencils: Free Multip. and Division templates showing a template for illustrating the 5 division strategies.  I completed two of them for you to see how these can be used effectively in the classroom.  They could also be used as anchor charts (and then student can create some of their own). There is also a companion set for multiplication strategies. Get your set here: Multip. and Div. strategy anchor charts

Enjoy your division journey!  Stay tuned for more!

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.  

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:

  1. When using manipulatives or drawings, instead of randomly placing the objects being divided into equal groups, arrange them so it’s easy for the child as well as the teacher to see at a glance how many there are. In other words, if there are 5 in each group, are they randomly scattered?  If they are, the child wastes a lot of time recounting, which often invites error.  And the teacher has to spend time rechecking the child as well. Or, are the objects arranged in smaller arrays or groups making it very easy to see the total (like dice? by twos?).  This little requirement adds to a child’s understanding of number bonds and multiplication.

  2. Instead of placing individual objects, have the students try tally marks.  Again, these are counted more efficiently than a random organization – and it aids in multiplication.
  3. Instead of always using a one-at-a-time strategy as objects are being distributed, help them think that often they can try 2 at at time, or 5 at a time.  This aids with estimation and helps transfer this idea to future long division processes – especially partial quotients.
  4. Connect use of manipulatives and drawings with real life stories. What things come in equal groups?

    Refer to one of my previous blog posts showing this template for stories about equal groups (which can be multiplication or division): Equal Groups blog post. Help students notice each problem consists of these three components.

    • # of groups
    • # in each group, and
    • the total #
    • The division problem will usually provide the total and one of these (# of groups; # in each group). So the problem will be to determine the missing component by relating known multiplication facts and/or dividing.

Stay tuned!  Next week I will include some helpful basic division concepts resources.

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?

Relate experience with arrays to determine area of a rectangle. This mostly just adds a measurement component to the problem.

  • Draw a rectangle and partition it into columns (length) and rows (width) to match the story.  Here are two sample stories:
    • I am making a rectangular shaped garden which I want to be 12 square yards in size.  If the length of the garden is 4 yards, how long does the side of the garden need to be?
    • I am using a rectangular piece of wood to cover a broken window that is 12 square feet.  One side of the wood is 3 feet. How long would the adjoining side be?

Here’s a great game called “Block it” which utilizes arrays. It can have variations depending on the level of your students. Here is a FREE copy of the directions: Block-It Game Directions

Materials needed:

  1. 1 grid paper (1/2″ is great)
  2. 2 players
  3. Each player needs 1 crayon or colored pencil (light colored). Different color per player.
  4. 2 number cubes or dice (6 sided).

How to play:

  1. Player 1 rolls the dice.  Let’s say a 3 and 4 are rolled. The player makes a 3 x 4 “block” or array. Be sure to show them how to use the lines on the grid paper to make this (as I discovered it’s not always clear to some students). Color it in with crayon.  Inside the block, write the product (12).
  2. Player 2 then rolls the dice and uses their 2 numbers to create another block, colors it, labels it, etc.
  3. Repeat
  4. The goal is to create as many blocks / arrays as possible (more than the opponent). There is a strategy to maximize the use of the space. Repeated play helps children see they need to consider this so they don’t end up with little unusable spaces.
  5. As the board gets filled up, players may have to miss a turn or roll again if not enough space is available.

Variations:

  1. As the board gets filled up, students may need to start thinking of alternate ways to make their blocks to fit the available space. For example, if the player rolls a 6 and 4 but there is no room to fit a 6 by 4 array, they can think of other ways to make an array of 24 that might work (such as 8 x 3, 12 x 2).
  2. Students can keep track of their score by keeping a running total of each block / array they make.
  3. Use smaller size grid paper and use 9, 10, or 12 sided dice.
  4. Write the fact family members for each block created.

Enjoy! Have you / your students played Block It?  Let us know if you like it!  

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.

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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)

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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.

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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.”

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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

Geometry Part 1: The Basics

by C. Elkins, OK Math and Reading Lady

For many schools, it seems as if Geometry and Measurement standards remain some of the lowest scored. This has always puzzled me because it’s the one area in math that is (or should be) the most hands-on — which is appealing and more motivating to students. Who doesn’t like creating with pattern blocks, making 2 and 3D shapes with various objects, using measurement tools, and getting the chance to leave your seat to explore all the classroom has to offer regarding these standards? So what is it about geometry and measurement that is stumping our students? Here are some of my thoughts – please feel free to comment and add your own:

  • Vocabulary? (segment, parallel, trapezoid, perpendicular, volume, area, perimeter, etc.)
  • Lack of practical experience? Not all homes have materials or provide opportunities for students to apply their knowledge (like blocks, Legos, measuring cups for cooking, tape measures for building, etc.).
  • Background knowledge about the size of actual objects? We take it for granted students know a giraffe is taller than a pickup truck. But if students have not had the chance to go to a zoo, then when they are presented a picture of the two objects they might not really know which is taller / shorter. Think of all of the examples of how we also expect students to know the relative weights of objects. Without background knowledge or experience, this could impede them regarding picture type assessments.
  • Standards keep getting pushed to lower grades when students may not have reached the conservation stage? If they think a tall slender container must hold more than a shorter container with a larger diameter, or they think a sphere of clay is less than the same size sphere flattened out, they may have difficulty with many of the geometry and measurement standards.

In this post, I will focus on Geometry. Here is a basic look at the geometry continuum (based on OK Stds.):

KG:  Recognize and sort basic 2D shapes (circle, square, rectangle, triangle). This includes composing larger shapes using smaller shapes (with an outline available).

1st:  Recognize, compose, and decompose 2D and 3D shapes. The new 2D shapes are hexagon and trapezoid. 3D shapes include cube, cone, cylinder, sphere.

2nd: Analyze attributes of 2D figures. Compose 2D shapes using triangles, squares, hexagons, trapezoids and rhombi. Recognize right angles and those larger or smaller than right angles.

3rd:  Sort 3D shapes based on attributes. Build 3D figures using cubes. Classify angles: acute, right, obtuse, straight.

4th:  Name, describe, classify and construct polygons and 3D figures.  New vocabulary includes points, lines, segments, rays, parallel, perpendicular, quadrilateral, parallelogram, and kite.

5th: Describe, classify, and draw representations of 2D and 3D figures. Vocabulary includes edge, face, and vertices. Specific triangles include equilateral, right, scalene, and isosceles.

Here are a couple of guides that might help you with definitions of the various 2D shapes. The 2D shapes guide is provided FREE here in a PDF courtesy of math-salamander.com.  I included a b/w version along with my colored version. The Quadrilateral flow chart I created will help you see that some shapes can have more than one name. Click on the link for a free copy (b/w and color) of the flow chart. Read below for more details about understanding the flow chart.

PLEASE note these very important concepts: Continue reading

All About 10: “Make a 10” and “Adding Up”

by C. Elkins, OK Math and Reading Lady

Last time I focused on some basics about learning the number bonds (combinations) of 10 as well as adding 10 to any number. Today I want to show the benefits of making a 10 when adding numbers with sums greater than 10 (such as 8 + 5).  Then I’ll show how to help students add up to apply that to addition and subtraction of larger numbers. I’ll model this using concrete and pictorial representations (which are both important before starting abstract forms).

Using a 10 Frame:

A ten frame is an excellent manipulative for students to experience ways to “Make a 10.” I am attaching a couple of videos I like to illustrate the point.

Let’s say the task is to add 8 + 5:

  • Model this process with your students using 2 ten frames.
  • Put 8 counters on one ten frame. (I love using 2-color counters.)
  • Put 5 counters (in another color) on the second ten frame.
  • Determine how many counters to move from one ten frame to the other to “make a 10.” In this example, I moved 2 to join the 8 to make a 10. That left 3 on the second ten frame. 10 + 3 = 13 (and 8 + 5 = 13).

The example below shows the same problem, but this time move 5 from the first ten frame to the second ten frame to “make a 10.” That left 3 on the first ten frame. 3 + 10 = 13 (and 8 + 5 = 13). Continue reading

All About 10: Fluency with addition and subtraction facts

by C. Elkins, OK Math and Reading Lady

I’m sure everyone would agree that learning the addition / subtraction facts associated with the number 10 are very important.  Or maybe you are thinking, aren’t they all important? Why single out 10? My feelings are that of all the basic facts, being fluent with 10 and the combinations that make 10 enable the user to apply more mental math strategies, especially when adding and subtracting larger numbers. Here are a few of my favorite activities to promote ten-ness! Check out the card trick videos below – great way to get kids attention, practice math, and give them something to practice at home. Continue reading

Comprehension: Point of View

by C. Elkins, OK Math and Reading Lady

Point of View seems to be a difficult skill for children to master. I have noticed it is high up on most schools’ lists of standards that need retaught and reviewed. So this made me wonder, “What is it about this skill that is being misunderstood?”

Here are my thoughts:

  • Part of it may be trying to determine “Which points of view are my students supposed to know?” In Oklahoma, the standards are fairly clear for grades 2-4 which emphasizes the ability to identify the first and third person points of view. But 5th grade isn’t as specific so many teachers are left wondering, “Do I include the 2nd person point of view? The Omniscient? . . .” (See a list below of the Pt. of View Stds. for each grade level. It appears they have clarified the 5th grade position since last year.)
  • Some of the confusion may be that students are mostly armed with the keywords regarding various points of view (1st = I, me, my; 2nd = you, your; 3rd = him, her, them, they). I have literally seen students counting pronouns and then declare the point of view based on which pronouns they saw the most of. This means they were not really focused on the overall jist of the story and/or were ignoring the fact that a quote using the word “I” doesn’t necessarily make the selection a first person point of view. This is where too much reliance on beautiful anchor charts on Pinterest can perhaps harm your students.  So be cautious!
  • Some of it may be that students confuse all of those words: Purpose, Point of View, Perspective.  Here is a good, short video from Smekenseducation.com which easily explains the difference. Click here to watch: Purpose, Point of View, and Perspective Video
  • Stay tuned for some cool FREE activities (end of post).

Continue reading

New Category List

by C. Elkins, OK Math and Reading Lady

I am pleased to be able to make my blog even easier to search for topics of interest to you! On the side bar you will now see an expanded category list with a number indicating how many posts I have written on that particular topic.

If you are looking at this on your phone, then the category list might appear at the bottom.

To get right to my FREE stuff, look at the black bar at the top and click on “Links to free downloads.” You will also see things arranged by math and literacy categories.

Enjoy!!

Comprehension Strategies (2nd-5th and above)

by C. Elkins – OK Math and Reading Lady

I have been doing some research about the difference between reading skills and reading strategies.  There seems to be a variance of opinions, but basically a reading skill is described as a path to answering certain kinds of questions (cause-effect, compare-contrast, sequence, etc.), while a strategy involves a higher meta-cognitive process which leads to deeper thinking about a text (visualize, question, summarize).  Another way to put it is this:  When reading, I need a strategy to help me understand when and where to apply the skills I have learned.

It probably can be illustrated more clearly using mathematics:  A skill might be adding two double-digit numbers, while different strategies might be these:  using base ten manipulatives, using an open number line, or the partial sums method.  Or soccer:  A skill would be the dribbling the ball (how to position the foot, how close/far to keep it to the player), while a strategy would be how to keep dribbling while keeping it away from the opposing team.

There are also varying opinions about which reading practices are considered strategies.  I like to think of strategies as those that can be applied to any reading text such as: summarize, visualize, question, make connections, predict, infer, author’s purpose & point of view. I need a strategy to help me understand when and where to apply the skills I have learned. Keep reading for more ideas and FREE resources.

Skills seem to be more dependent on the text structure (meaning they only apply to certain texts) such as sequence, compare/contrast, cause/effect, main idea / detail, problem-solution, identify story elements, etc. 

  • To help me visualize (strategy), I might use skills about character analysis such as paying attention to their words and actions to help me “see” what is really going on. Another example:  I might use skills about noting details while reading a passage to make the details “come alive” as I try to picture them in my mind. (See link to strategy posters at the end of this post.)

To help me summarize (strategy) an article, I need to analyze the text structure (skill) and then use that information to help me summarize. 

  • Is it in sequence? Then my summary will use words such as first, then, next, last.
  • Is it comparing and contrasting something? Then my summary will need to use words such as alike or different.
  • Is it informational? Then my summary will list facts or details.
  • Is it fictional? Then my summary will tell the characters, setting, and events.

Continue reading