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.”
Ideas:
Misconception #2: A student hears this: “To find area, multiply the length times the width.”
Ideas:
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.
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.
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.
Next post: Ideas to address Misconceptions #3, 4, and 5 and links for area and perimeter activities!! STAY TUNED
]]>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:
Acute Angles (angles less than 90°):
Obtuse Angles (angles greater than 90°):
Lines: Parallel, Perpendicular, Intersecting
Some good hands-on activities:
Share some of your favorite lines and angles teaching points or activities!!
]]>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.
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).
In my opinion, this hands-on experience really enables students to see a way to calculate the surface area is:
Go to illuminations.nctm.org activity called “Cubes”. It is an interactive lesson in which you can change rectangular prism nets to your specs and see how surface area and volume are determined. There are lots of cool activities at this site (nctm = National Council for Teachers of Mathematics).
Enjoy!! Next time I will focus on another aspect of geometry — angles and lines.
]]>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.
3. Give students these various quadrilaterals: square, rectangle, parallelogram, trapezoid
The activity sounds similar to the above, but students will be using their knowledge of shape attributes and properties (Level 2 or above) to decompose them into smaller shapes. and name / describe the decomposed shapes. More advanced students can come up with ways to compute the area based on these decompositions and then relate them to the standard formulas.
4. Use tangrams to create random designs, shapes, or other recognizable pictures. Most tangram sets come with some outlines to follow. In case you don’t have any, here are some good resources.
Still more composing and decomposing to go, especially with 3D shapes. STAY TUNED!!!
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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?
2. How many rectangles can you make using 2 or more squares? (Level 0-1)
3. How many different ways can you make a rectangle using 12 tiles? 24 tiles? Record on graph paper. (Level 1 or 2)
4. Make a rectangle using 24 tiles. If you decompose it, how many different smaller rectangles or squares can you make? Explore with tiles first, then record results on graph paper.
5. How can you use color tiles to decompose a polygon into smaller rectangular shapes? For students to master this skill, they must have a good understanding that opposite sides of rectangles are equal length. This is also crucial for determining area.
Here are some resources using one inch tiles:
Pattern Blocks:
Put pattern blocks together to create another shape:
This encompasses composing (because they are using smaller pieces to build a larger shape), but also decomposing if the student shows multiple ways to fill in a shape — 2 trapezoids = 1 hexagon; 3 rhombus = 1 hexagon; 6 triangles = 1 hexagon. OR, 1 hexagon = 1 trapezoid and 3 triangles; 1 hexagon = 1 trapezoid, 1 rhombus, and 1 triangle, etc. This exploration is also beneficial to understanding equivalent fractions.
Here are some resources using pattern blocks:
Do you have any ideas to share about composing / decomposing using tiles or pattern blocks? More on composing / decomposing next time (Tangrams, etc.)
]]>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.
Level 0 Visualization: Describe shapes on the basis of their appearance.
Level 1 Analysis: Describe shapes on the basis of their properties.
Level 2 Abstraction: Recognize the relationships between the properties of geometric shapes.
The following 2 levels usually encompass thinking involved in high school, so I will not provide more details on these in this post.
Level 3 Deduction: Move from thinking about properties to reasoning or proving theories.
Level 4 Rigor: Establish and analyze theorems in different postulation systems.
Geometry Resources
For those of you interested in more of van Hiele’s theory and specific ways to apply to different topics within Geometry, here is another article: van Hiele Model of Geometric Thinking (Ohio Dept. of Education)
I know this post doesn’t focus too much on lesson ideas – BUT it is a very important regarding developmental levels of geometric understanding. More lesson ideas to come.
]]>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:
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:
*Adjacent means the sides share a common vertex. You can see that to recognize the attributes and adequately describe, compare, draw, or construct these shapes, students need to have a very good understanding of types of angles and parallel lines.
Another geometry term that needs to be understood is POLYGON:
A polygon is a closed 2D figures with straight sides. It includes all of the 2D shapes on the above guide (but does not include the circle – because it has curves). If the shape had curves or doesn’t fully connect, then it can’t be called a polygon. A regular polygon is one in which the sides are all the same length and whose angles are all the same. So any 8-sides polygon would be called an octagon, but a regular octagon would have 8 equal sides and angles. Check below for some FREE resources about identifying polygons.
Next week: I will go more in depth regarding some of the geometry standards above to provide hands-on ideas and help clear up student misconceptions. Enjoy!
]]>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:
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).
The making 10 strategy is also really helpful when adding 9. For example: 9 + 7 — We tell students to think 10 + 6, but showing them with the ten frame makes it less abstract when first learning.
Decomposing Numbers:
This is a pictorial representation of the same problems showing how to decompose one of the addends to make a ten.
Here are 2 good resources for this method:
Adding Up:
I LOVE, LOVE, LOVE using the add up strategy for most double+ digit subtraction problems. Making a 10 is the essential first step. There are a few ways to show this. I’ll start with a concrete method first using place value disks.
The problem is 100 – 37.
Here’s the same problem using “The arrow way” from Eureka / Engage NY.
Open Number Line:
Here’s the same problem using an open number line, which is a great pictorial model showing the process. Using an open number line to count up / add up for a subtraction problem should help them see “the difference” between the two numbers.
See how this strategy also works well with a missing addend problem: 3,249 + n = 6,500
Answer (add the bold # above): n = 3,251
Now I’m not suggesting that we should ignore the standard algorithm, but using the make a 10 strategy is very helpful when doing mental math. It’s a very helpful strategy when paper and pencil or a calculator are not available. It’s also a helpful strategy for children to check their work. Instead of students asking, “Did I get this right?” I tell them to try a different strategy — if they get the same answer they are probably correct.
I have often asked students this question to see what strategy they use: “What is 100 – 88?” So many go right away to the std. algorithm – crossing out zeros, etc. when the adding up strategy would have been more efficient (88 + 2 = 90 and 90 + 10 = 100)!!
Enjoy! Let us know your favorite “ten” lesson or activity.
]]>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.
Hold a card with “10” in one hand and another card such as “4” in the other hand. With the cards facing the students, slide the 4 over top of the zero in 10. The result of course is “14.” For some reason students really think of it as a cool trick. Be sure the one’s digit is written close to the left edge of the paper so when you slide it over the 0 in “10” it won’t cover up the ten’s place. Try the same thing with 20 + 4 and other combinations involving multiples of 10 combined with ones. Then students should be able to do it mentally.
This is a tic-tac-toe game involving numbers. In this game, a student wins if he/she can complete 3 in a row using 3 numbers which combine for a total of 10. So this one is a little higher level thinking – but a lot of fun! Use a dice or number cube. Decide where to put the number rolled. Students each use a different colored crayon or marker to enter their numbers onto the grid. The strategic part comes when trying to determine the different scenarios possible depending on what you roll. (Get it via “4 Games About Ten” above.)
When viewing the second video (the actual trick with no explanation), you might wonder why at the end I was able to correctly guess the mystery card was a 9 (when the other card next to it was a 4). This is because as I was matching up the piles of cards at the end, I already had noticed the Ace card (1) had no match for it, so I just scooped up the remaining cards and the 4 just happened to be on top. So sorry if I confused you. This was my first attempt at posting a video on my blog. I’m learning!!
Next post I will explore more about how knowing the basic facts of 10 mentioned above will help with other facts as well as adding or subtracting larger numbers. What are your favorite facts of 10 activities?
Enjoy!
]]>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:
A lot of us love using “The True Story of the Three Little Pigs” by Jon Scieszka. Here’s how these three P’s work out for this story.
I was so concerned about this standard that I asked the OK ELA Elementary director for some advice. She told me that yes, students need to be able to identify the point of view . . . But students (especially in 3rd – 5th) need to be able to go deeper to determine how the point of view affects the information presented in the selection: Is it one-sided? Is it a more global view? Is there just one or multiple points of view? Does the point of view used give more insights into the character or the situation? Just one character or all of them?. I thought these were all very valid points! She also said the ELA Frameworks committee of teachers would be updating information on their website and it appears they have now clarified it, especially for 5th grade (in Oklahoma). See the link below.
So this tells me I have to be super cautious about how I present this information to students.
Here is some very helpful information from the 5th grade OK ELA Point of View std.
Here is the 4th grade OK ELA Point of View std.
Here is the 3rd grade OK ELA Point of View std.
Here is the 2nd grade OK ELA Point of View std.
Here is the 1st grade OK ELA Point of View std.
I have selected some FREE resources I think are worthy of using. If you have some you think are great to share, please let us know!!
Enjoy!! Let us know of your experience with point of view vs. perspective.
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