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.”

Misconception #2:  A student hears this: “To find area, multiply the length times the width.”

  • Problem:  Student doesn’t know their multiplication facts.
  • Problem:  Student doesn’t know which dimensions are the length and width.

Ideas:

  • A good guide:  The length is the longest side. The width is the shortest side.
  • The length and width are two adjoining sides (not opposite sides).
  • Show how to partition a rectangle into squares.  If a rectangle had a dimension of 4″ x 2″, then show them how to make 4 columns and 2 rows.  Watch how they do this.  For many students, they would draw 4 vertical lines inside to make the columns and 2 horizontal lines to make the rows.  This would obviously result in a 5 x 3 set of squares – but students don’t always check.  Do they know — “To divide a rectangle into 4 columns, I only need to draw 3 lines.” Now, counting squares isn’t the most efficient method – but one that might help students who struggle with the concept or with multiplication facts. 
  • Some rectangles are too large to draw squares in. If students don’t know their multiplication facts, the rectangle can be partitioned into 2 (or more) smaller rectangles using facts they do know.  Then the area of each smaller rectangle is added together to find the total area. See picture below for an example.  Again, students have to connect to geometry to understand that opposite sides of a rectangle are equal.
    • Example:  A rectangle has dimensions of 8 x 7.  Using facts more readily known, break one of the dimensions into 2 addends (such as breaking 7 apart into 5 + 2). Partition the rectangle into 2 rectangles and use the 5 + 2 to label one side (instead of 7).  Using the concept of the distributive property, the student is calculating this:  8 x 7 = 8 (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 sq. units. 

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.

Misconception #4:  A student hears this:  “Record your measurement for area as square inches and the measurement for perimeter as inches.”  Note: This applies to use of units such as cm, feet, meters, yards, miles, and so on.

  • Problem:  The student doesn’t understand the difference between square and non-square measurements.

Misconception #5:  Students think there may be a relationship between area and perimeter. They may think all shapes with the same area have the same perimeter.

  • Problem:  This means if one shape has an area of 12 square inches, and the perimeter is 16 inches, they might think all shapes with an area of 12 have perimeters of 16 inches.

Next post:  Ideas to address Misconceptions #3, 4, and 5 and links for area and perimeter activities!!     STAY TUNED

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°):

  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 ray is wider than the paper corner, it is >90°.
  2. Students are told an obtuse angle is big, and may classify an angle with long rays as obtuse (because it is a large picture).
    • Remind them it’s the size of the angle that makes it obtuse, not the size of the lines.
  3. While right angles are exactly 90°, students may expect an obtuse angle to be given a specific number.
    • Obtuse angles range from 91° to 179°.

Lines: Parallel, Perpendicular, Intersecting

  1. Students get the terms parallel and perpendicular confused.
    • Notice the word “parallel” has a set of parallel lines in the word (the two l’s).
  2. Students may see two lines and assume they are parallel just because they are not intersecting.
    • Remind them that parallel lines are those who are the same distance apart.
    • Show that 2 non-intersecting, non-parallel lines would intersect if  you extended their lines.
  3. Students think lines must be intersecting OR perpendicular.
    • Show them that all perpendicular lines are also intersecting lines, but not all intersecting lines are perpendicular.
    • Perpendicular lines are a special type of intersecting line (with 90° angles).

Some good hands-on activities:

  • Locate angles and lines of all types in the classroom.
  • Bring in pictures and/or photographs of different lines and angles for collages or posters. Architectural features would be an excellent source.
  • Create them with toothpicks, chenille stems, craft sticks, paper strips with brads, etc.  BUT, help students see that acute and obtuse angles can be turned different directions and come in different sizes (acute angles range from 1° to 89° and obtuse range from 91° to 179°).
  • Using an analog clock, name times in which the hour and minute hands form acute angles, right angles, obtuse angles, and straight angles. Here are some examples:
    • Acute angles:  1:00, 2:00, 6:20
    • Obtuse angles:  4:00, 8:00, 1:30
    • Right angles:  3:00, 9:00, 12:15 (approximately)
    • Straight angles:  6:00, 12:30 (approximately — or 12:32 exactly due to hour hand placement)
  • Use a map of the United States to classify states according to the type of angles in their borders.
    • Which states have right angles?
    • Which states have acute angles?
    • Which states have obtuse angles?
    • Do any states have all 3 types?
  • Using the capital letters of the printed standard alphabet, classify them by the types of lines and angles. Provide individual copies of the printed alphabet to each student (or pair of students). Click here for a FREE recording sheet:  Classify Letters of the Alphabet CE:
  • Here is a FREE measuring angles partner game called “Tangling with Angles .”  A student draws an angle. Partner names the type of angle and estimates the measurement.  Points are earned based on the difference between the estimate and the actual measurement. The partner with the least number of points is the winner.
  • This is a good set of posters by Kathleen and Mande Lines and Angles Posters $2.25 @ TPT

Share some of your favorite lines and angles teaching points or activities!!

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

Math Problem Solving Part 5: Multiplication and Division Comparisons

by C. Elkins, OK Math and Reading Lady

As I promised, here is a post about another multiplication and division story structure.

The dinosaur is about twice as tall as the human. The dinosaur is about 12 times wider than the human.

While the previous structure I mentioned dealt with equal groups, this one deals with comparisons.  Like with addition / subtraction comparison problems, I will show some ways to use double bars to illustrate these. (I just added a FREEBIE – a 2-pg. pdf showing the pictures of the problems shown below – get it now Equal groups comparison problem pics)

Notice some of the questions associated with these stories. They sound a lot like questions used in other story structures — this is why I advise against statements like, “If it asks how many in all, add the numbers together.” Notice how this method guides a student as they go from pictorial to abstract. You can also represent these types of problems with cubes or money manipulatives if you need a concrete example.

This problem shows a multiplying process.

  1. Read the problem. Ask who and what this story is about (Joe and Brent – Joe has more $ than Brent). Notice  there is no additive or subtractive process, but one has more – the other has less. This is a signal to use comparison bars to help solve.
  2. Make a vertical line (to help line up the bars evenly on the left side. Draw the bar with the known amount. (Joe has $22.)  Label the second bar (Brent).

3. Since the story said Brent has 3 time more, draw 3 bars equal to the size of Joe’s bars.

4. Since Joe’s bar is $22 and the bars by Brent’s name are equal in size, all of Brent’s bars should be labeled with $22.

5.  To find out how much Brent has, solve by repeated addition or multiplication.

6.  IF the question was slightly different (How much do Joe and Brent have all together?), then the above steps would have to be followed by 1 more step (adding the two amounts together).

* If students had been taught earlier to “Add when the question asks How many in all” then the child would likely add $22 + 3 to get their answer.  That signals no conceptual understanding of what the problem is all about:  One has more, the other has less.

The following example involves the division of a bar to help solve it. Continue reading

Math Problem Solving Part 4: Equal Groups

by C. Elkins, OK Math and Reading Lady

Are you ready for a way to help students solve multiplication and division word problems? I have developed a template that will help students record the given facts and think through the process involving equal groups. There are 3 basic types of equal groups problems:

  • # of groups and # in each group are known
  • # of groups and total are known
  • # in each group and total are known

Pictured here is an illustration of the strategy which can be used for multiplication and division problems. Get my FREE packet right here: Equal groups strategy with template.

One of the most important steps I recommend when working with equal groups problems is for students to brainstorm things that typically come in equal groups. There is a great book titled “What Comes in 2’s, 3’s, and 4’s” which can set the stage. It’s a picture book, but helpful to get kids’ brains warmed up.  Here are some visuals I made to illustrate the point. Click here for your FREE copy. They are included in this set of Equal groups pictures and list template

Kids may need your help to think of things to add to the list. See some hints below. This template and a full list is included in the above FREE download. Continue reading

Math Problem Solving Part 3: Comparing problems

by C. Elkins, OK Math and Reading Lady

Solving comparison problems involves a different thinking process than addition or subtraction problems. In my previous posts Math Problem Solving Parts 1 and 2, I focused on addition and subtraction problems in which I advised students to think of the verbs used (found, bought, earned, spent, lost, gave away, etc.) to help visualize what is going on in the story.  In comparison problems, there are no additive or subtractive processes present – just analyzing which has more, which has less, and how much more or less one amount is compared to another.

I love using a double bar method for comparison problems. It can be represented with manipulatives for a concrete experience (ex: connecting cubes, tiles) or with a drawing of two rectangular bars for a pictorial representation. These two visual models provide students with the concept of comparison — they can see right away which has more, which has less. Then the information can be used to compare the two amounts. Look for the FREE resource at the end of this post.

With manipulatives:

I recommend the use of manipulatives when dealing with students’ first experience with comparison problems. Dealing with quantities less than 20 make using manipulatives manageable. I will show this in a horizontal format, but it can certainly be done vertically. You can also apply this quite well to graphing problems.

Problem:  I have 12 crayons. My friend has 8 crayons. How many more crayons do I have than my friend?

  1. Determine that this problem is not an addition / subtraction process (because no one is adding to what they have and no one is giving away what they have), but in fact a comparison problem.
  2. Be cautious about focusing on the question”how many more” and telling students that when they see this they need to subtract. When we tell kids to focus on the specific words in a question with a “rule” for adding or subtracting, they start to lose sight of the actual story.
    • Consider that this question can also be used with a SSM (some and some more) story which is not a comparison problem:  What if it read: “I have 8 nickels. I want to buy a candy bar that costs 80 cents. How much more money do I need?” This story means: I have some (40 cents) and I need some more (?) so that I have a total of 80 cents. This is a missing addend or change unknown story: 40 + ___ = 80.  Even though it could be solved in a similar manner as a comparison story, we want students to be able to tell the difference in the types of stories they are solving.
  3. Determine who has more (represented by yellow tiles), who has less (green tiles).
  4. The crayon problem can be solved by lining up 2 rows of manipulatives (pictured):
    • Notice the extras from the longer bar. Count them (4)., or
    • Count up from 8 to 12 to find the difference.
    • Even if the question was “How many fewer crayons does my friend have?” it would be solved the same way.

With pictorial double bars:

Problem Type 1 (Both totals known):  Team A scored 85 points. Team B scored 68 points. How many more points did Team A score than Team B?

  1. Determine this problem is not an addition / subtraction process (because the teams are not gaining or losing points).
  2. Ask “Who?” and “What?” this story is about:  Team A and B and their scores.
  3. Draw double bars (one longer, one shorter) which line up together on the left side.
  4. Label each bar (Team A, Team B).
  5. For the team with the larger amount (Team A), place the total outside the bar (85).
  6. For the team with the smaller amount (Team B), place the total inside the bar (68).
  7. Make a dotted line which extends from the end of the shorter bar upward into the longer bar.
  8. Put a ? inside the extended part of the longer bar. This is what you are trying to find.
  9. To solve, there are 2 choices:
    • 68 + ____ = 85     This choice might be preferred for those with experience using mental math or open number lines to count up.
    • 85 – 68 = _____

Continue reading

Math Problem Solving Part 2: Separate (aka Some, Some Went Away)

by C. Elkins, OK Math and Reading Lady

In the previous post, I addressed problems dealing with an additive process (join; aka SSM).  In this post, I will show you some models to use for these types of problems:  Separate; aka Some, Some Went Away — SSWA.  I will share some models which are great for KG stories as well as templates that are helpful for intermediate students to use, especially when dealing with missing addend types.

As I previously mentioned, it is my belief that students should focus more on the verb / action in the story and not so much with the key words we often tell kids to pay attention to. Brainstorm actions that signify a subtractive process.  Post it in the class.  Keep adding to it as more actions are discovered. With this subtractive action, kids should quickly realize that there should be less than we started with when we take something away. Here is a FREE poster showing some of the most common subtraction action verbs. Click HERE to get your copy.

Some of the work mats pictures below come from the following source. These are great for KG-2nd subtraction storytelling.   Subtraction Pack: A Pinch of Kinder by Yukari Naka

Like with all story problems, I model how I reread the problem several times.

  • First read — Just read it
  • Second read –Identify who and what the story is about (the action).
  • Third read — Decide what to do with the numbers. Is a given number the wholetotal amount or part of the amount? Do I know how the story started? How it changed? The result?

Here are 3 types of subtraction story structures: Continue reading

Math Problem Solving Part 1: Join (aka Some and Some More)

by C. Elkins, OK Math and Reading Lady

I have 1 penny in one pocket. I have 6 more pennies in another pocket. This is a Join or “some and some more” story structure.

Many teachers I work with have asked for advice on problem solving in math. Students often don’t know how to approach them or know what operation to use. Should teachers help students focus on key words or not? What about the strategies such as CUBES, draw pictures, make a list, guess and check, work backwards, find a pattern?

While all of those strategies definitely have their purpose, I find  we often give kids so many steps to follow (underline this, circle this, highlight that, etc.) that they lose sight of what the problem is basically about.

In this post, I will focus on two basic questions (who and what) and a simple graphic organizer that will help students think about (and visualize) the actions in a one-step Join story problem. KG and first grade students can act out these actions using story mats or ten frames. Late first through 5th grade can use a part-part-whole box. There are two FREE items offered.  

These are the types of problems I will focus on in the next few posts.

  1. Join (also referred to as SSM – Some and Some More)
  2. Separate (also referred to as SSWA – Some, Some Went Away)
  3. Part-Part-Whole
  4. Comparing
  5. Equal groups

JOIN problems have 3 versions: 

  • a + b = ___     (The result is unknown.)
  • a + ____ = c   (How the story changed is unknown / missing addend.)
  •  ____ + b = c  (The start is unknown / missing addend.)

They can also be referred to as “Some and Some More” stories (SSM). This means, you have some and you get some more for a total amount. These present themselves as additive stories, but it doesn’t necessarily mean you have to add to solve them. The second and third version above are often referred to as missing addend problems. Continue reading

Math Meetings KG-3rd Grades

by C. Elkins, OK Math and Reading Lady

For my Lawton, OK friends who are implementing Saxon Math but don’t have wall space for all of the math meeting components, here are a few links from TPT for grades KG-3rd for ppt / SMARTBOARD versions.  The important aspect of the morning meeting is the interactive part where individual students have a role in placing the numbers, days, coins, patterns, clock hands, graph piece, etc. on the board each day. With that said, there is also value in having some of the math meeting components visible throughout the day such as the calendar, 100 chart, and number of the day. Think of the components you or your students would most likely refer to outside of the math meeting time to keep a permanent physical display.

Here are the links. Read the other purchasers’ comments and look at the previews to get more info.  I have not purchased any of these so I can not vouch for the quality or usefulness. For those of you who already have a math meeting ppt that you recommend, please let us know!! Thanks!

KG-2nd:

1st Grade:

2nd Grade:

3rd Grade:

Back to School

by C. Elkins, OK Math and Reading Lady

Here are a few links to some previous posts regarding literacy and math you might be interested in to help you start your journey this year.  And in case you didn’t see it, I now have an easy link to most of my own free resources. Creating this was one of my summer projects. Click here to get it now, but it is also available in the black bar above. See info at the end for another $25 subscriber / comment challenge!!  Have a great start to your year and Enjoy!!!

  1. Getting to know you literature connection and math activity
  2. Building a classroom community (includes link to great team building practices)
  3. Writing part 1
  4. Guided Reading Part 1: Getting Started
  5. Guided Reading Part 2: Routines and Procedures
  6. Meaningful Student Engagement: Whole Class Reading
  7. Daily Math Meeting Part 1: Building Number Sense
  8. Daily Math Meeting Part 2: Subitizing
  9. Addition and Subtraction Part 1: Numerical Fluency
  10. Addition and Subtraction Part 3: Facts Strategies
  11. Multiplication Strategies Part 1
  12. Fractions Part 1: The basics

 

Math Art Part 2: Decomposing and composing squares and triangles

by C. Elkins, OK Math and Reading Lady

I wanted to show you another example of math art, this time using squares and triangles. This project also falls under the standards dealing with decomposing and composing shapes. With this project, students can create some unique designs while learning about squares, triangles, symmetry, fractions, and elements of art such as color and design. It would be a great project for first grade (using 2 squares) or for higher grades using 3 to 4 squares.

A great literature connection to this project is the book “The Greedy Triangle” by Marilyn Burns. (Click link to connect to Amazon.) The triangle in this book isn’t content with being 3-sided and transforms himself into other shapes (with the help of the Shapeshifter). Lots of great pictures showing real objects in the shape of triangles, squares, pentagons, hexagons, and more.

Marilyn Burns is a great math educator to check out, if you haven’t already. She has a company called Math Solutions (check out MathSolutions.com). Marilyn and her consultants have wonderful resources and advocate for constructivist views regarding math education. She is also the author of Number Talks and many math and literature lesson ideas.

The 4 Triangle Investigation

Materials needed:

  • Pre-cut squares 3″, 4″ or 5″ (I used brightly colored cardstock.)
  • Scissors and glue
  • Background paper to glue shapes to

Directions

  1. Model how to cut a square in half (diagonally) to make two right triangles. (I advocate folding it first so that the two resulting triangles are as equivalent as possible.)
  2. Guide students into showing different ways to put two triangles together to form another shape. Rule: Sides touching each other must be the same length. Let students practice making these shapes on their desk top (no gluing needed). 
  3. Help students realize they may need to use these actions:
    • Slide the shape into place
    • Flip it over to get a mirror image
    • Rotate it around in a circular motion to align the edges
  4. Students are then given 2 squares (to be cut into 4 triangles) and investigate different shapes they can make following the above rule. Here are some possibilities:
  5. As the teacher,  you can decide how many creations you want each student to attempt.
  6. These shapes can be glued onto construction paper (and cut out if desired).
  7. As an extension, shapes can be sorted according to various attributes:
    • # of sides
    • symmetry
    • # of angles
    • regular polygons vs. irregular

Continue reading

Math Art Part 1: Fraction circle art (3rd-5th)

by C. Elkins, OK Math and Reading Lady

Incorporating art with other subjects is a great way to engage students. In this post I will share one project which helps students gain hands-on experience with fractions. More to come in future posts.

Fraction Circle Art

This project is inspired by Ed Emberley’s book “Picture Pie.” This is my favorite of his collection in which he shows dozens of ways to use fractions of circles to create almost anything. This book features mostly animals, flowers, and geometric designs. Students start with a circle (pre-cut with a circle cutting press is best, but you can also make nice circles by tracing around a can or drinking glass and cutting them out). Then the circle is folded and cut into these different fractional parts to create the design: halves, fourths, eighths, and sixteenths. These pieces are manuevered (think translations!!) and combined to make the desired art.

The pictured creations were made by 3rd and 4th graders during a session I conducted with them at Eisenhower Elementary. Here are a few of them.  So nice!!

Continue reading

Fractions Part 6: Adding and Subtracting Fractions

by C. Elkins, OK Math and Reading Lady

Starting in 3rd grade, students start building understanding about adding and subtracting fractions by composing and decomposing simple fractions using concrete and visual models. Composing: 1/4 and 3/4 combine to make 4/4 (whole). Decomposing: 8/8 is made up of 2/8 and 6/8. In fourth grade, students begin to add and subtract fractions with like denominators, but should still be utilizing models, drawings, and number lines to illustrate and simplify. In fifth grade, students are expected to add and subtract fractions of all types (proper, improper, with unlike denominators, etc.).

With a firm foundation of composing and decomposing, partitioning, comparing, naming equivalent fractions, and understanding the relationship between certain fractions (such as halves / fourths / eighths / sixteenths; and thirds / ninths / sixths / twelfths; or fifths / tenths), then students are more prepared to perform operations with fractions. Here’s a great resource by Donna Boucher at Math Coach’s Corner: Composing and Decomposing Fractions activity on TPT ($6)

Estimating: This is an important part of operations with fractions. Do you expect your answer to be less than 1/2, more than 1/2, more than 1? How do you know? If I was adding 8/9 + 11/12, my answer should be about _____? It should be slightly less than 2 because both of these fractions are almost 1.

If I am adding 4/6 and 6/8, my answer should be more than 1 because each of these fractions are greater than 1/2.

Different strategies: There are many “tricks” or shortcuts available to show students how to quickly add, subtract, or multiply fractions. I believe these shortcuts are only useful after a students has a strong understanding of why and how to find a common denominator and equivalent fraction. These shortcuts do not help build conceptual understanding of fractions.  I will focus on ways to understand the why using visual and pictorial models. Get your FREE copy of the following guides by clicking HERE. Continue reading

Fractions Part 5: Equivalent Fractions

by C. Elkins, OK Math and Reading Lady

This is part 5 of a series of fractions posts. Thanks for sticking around! Through explorations with fraction manipulatives, pictures, and drawings, we hope students begin to notice there may be different ways to express the same area using fractional terms. To cut a sandwich into halves and eat one of the halves is the same as cutting the same sandwich into fourths and eating two of the fourths. Read on for several freebies about equivalent fractions.

Students can gain experience finding equivalent fractions using models in several ways: Fraction strips, area models, set models, bar / length models, and number lines. Then with a strong understanding using concrete and pictorial models, the student is ready to apply paper-pencil methods to name equivalent fractions. But remember to use the same size whole: As in this picture using pattern blocks, 2/3 of a trapezoid is NOT the same as 2/3 of a hexagon.

Fraction Strips: If you don’t have sets of fraction strips, here is a free resource Fraction, decimal and percent strips charts via Kim Tran (TPT).  Commercially available strips are also nice. But, with some 1″ strips of construction paper, students can create their own and probably learn a lot about the relationship between halves, fourths, and eighths as well as thirds, sixths, and twelfths in the process of partitioning and cutting them.

TIP:  Make sure students know how to read a fraction strip chart. A couple of years ago I was working with a third grade class and assumed they could readily see that 5/10 was equivalent to 1/2. But after confusing looks, I realized that I needed to physically show them how to follow a line vertically down the chart to find other fractions that were in line (by placing a ruler or long pencil along the vertical line). Another example.  “To find another fraction equivalent to 1/3, find the line at the end of the 1/3 section and trace it vertically down the page to see if there are other fractions that stop along that same line. You should see in the sixth’s line that 2/6 lines up, and in the ninth’s line that 3/9 lines up and in the twelfths line that 4/12 lines up.”

After cutting and labeling strips, then explore equivalent fractions (those with the same size length). Students should be able to generalize that different fractions can used to represent the same area.

Area models:  By covering or partitioning shapes, students should notice that even though the same area is covered or shaded,  the number and size of the parts can change. Continue reading