Virtual math tools

by C. Elkins, OK Math and Reading Lady

Every once in a while you come across something wonderful, and you want to share with your friends.  Well, I am doing that with this FREE website.  It is https://www.mathlearningcenter.org/resources/apps

Here is what you will find.  Click the i on each app and you get great visual instructions about the tool bar at the bottom of each app.  These can be used on your Smartboard as well as installed as an app on a laptop or ipad, etc. A few of the apps have a share / copy feature (a box with an arrow coming out). All of them have a writing tool to accompany the app.

  • Fractions: Fraction bars or circles
  • Geoboard:  3 different boards, put stretchy bands on (no more worries about breaking them with this app), use for area, perimeter, shapes, arrays, area of irregular shapes
  • Clock: Program the hands and the clock (Roman numerals, minute guide), shade parts of the clock, show elapsed time
  • Math Vocabulary Cards:  Great for review or quiz. Adjustable for different math topics and grade level. 3 parts on each review question:  Term, definition, picture
  • Money Pieces:  Display and hide coins.  The coins can be shown as part of a block to relate to base ten blocks. The coins do seem a little small in size, however.
  • Number Frames:  5, 10, and 20 frames, 100 grid, counters, and objects.  The 100 grid can be adjusted to make any size array (up to 10 x 10).
  • Number Line:  Use for skip counting, addition, subtraction, fractions
  • Number Pieces:  This includes base ten pieces. These can also be used to show the area model for multiplication.
  • Number Rack (aka Rekenrek):  A great tool for primary grades. Based on use of 5 and 10 as benchmark amounts. Use 1-10 Rekenreks. Count by 5’s, Count by 10’s. Practice sliding the beads – it’s fun!  Here is a link from my blog on ways to use a Rekenrek:
  • Pattern Shapes (Blocks): Compose and decompose shapes. Create using the blocks: Duplicate, rotate, change colors! The sillouette shapes enable you / students to use blocks to fill in.  Plus for intermediate grades:  There is an angle measure tool. Measure angles of the polygons presented.
  • Partial Products Finder:  Make arrays. Slide the bar on the bottom or side to partition the rectangle into smaller parts. Tap on a section to see a different color.

I will add this link to my instructional resources for future reference.  Enjoy!

I’ll get back to phonics next time.  Have a great week!

Discovering Decimals Part 2: Addition & Subtraction

by C. Elkins, OK Math and Reading Lady

Last week, we looked at some ways to gain number sense about decimals. This post will address using decimals in the operations of addition and subtraction . . . and how to model concretely and pictorially. You can also download the color grid pages along with a free decimal math game in this post. Part 3 (future post) will address multiplication and division of decimals.

If you missed last week’s post, please review it first before continuing with this one. Before performing various operations with decimals, students must have a basic understanding of how to represent them concretely, pictorially and numerically.  Example:  .8 = .80 can be proven with base ten blocks and with 100 grid drawings. This understanding should also be linked to fractions: 8/10 is equivalent to 80/100. Click here for pdf of Representing Decimals page.

Addition  

For concrete practice, use a 100 base ten block to represent the whole (ones), the tens rod to represent tenths, and unit blocks to represent hundredths. Construct each addend and then combine them. Ten tenths’ rods become one whole. Ten hundredths cubes become one tenth.

In a pictorial model, shade in the ones, tenths and hundreds on hundred grids. Use different colors to represent each addend. Click here for pdf of Adding Decimals page.

The concrete and pictorial models will also prove .8 = .80, reinforcing the concept that adding a zero to the right of a decimal does not change its value. This will be an important factor when moving to the standard algorithm vertical addition model.

Using an open number line is also a good pictorial model to use when adding decimals, especially if students are already familiar with its use regarding addition of whole numbers. This method reinforces number sense of the base ten system because you continually think, “What goes with .07 to make a tenth?” (answer = .03); or “What goes with .9 to make a whole?” (answer: .1).

A method called partial sums should also help students gain place value number sense along with addition of decimals. Each step is broken down (decomposed). 

Estimating will also be a critical addition problem solving step focusing on number sense. If adding 34.78 plus 24.12, does the student realize their answer should be somewhere close to 35 + 24? Are they thinking, “Is my first number closer to 34 or 35?”  What is halfway between 34 and 35 (34.5 or 34.50)? Where would 34.78 fall if it was on a numberline between 34 and 35? (See my post on rounding for more details.) Continue reading

Decimals: Part 1 – The Basics (revised)

by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes. Continue reading

Multiplication facts: What happens when students don’t or can’t memorize them?

by C. Elkins, OK Math and Reading Lady

If you teach 3rd and above, I am positive you have students who have not memorized their multiplication facts. So what do they do to try to get the answer? From my experience, most students seem to know that repeated addition, drawing equal groups or arrays, and skip counting are strategies to try. I do believe those are very helpful for students to conceptualize what multiplication is all about. But here is what is frustrating:

Let’s say the problem is 6 x 7:

  • Do they write 7 + 7 + 7 + 7 + 7 + 7 and then add each part at a time? Or a little better, do they add 7 + 7 three times?
  • Do they draw a picture such as 6 circles with 7 items inside each one? The main difficulty with this is most students using this method count each object one at a time, making this a counting practice, not multiplication practice.
  • Do they draw an array? If so, do they correctly line up the rows and columns? Do they count each item in the array one at a time? Or do they group some together (which is a little better because they are at least thinking of equal groups)?
  • Do they skip count by fingers or write the sequence on paper? And what happens then? They may start off okay with 7, 14, 21 and then repeatedly count 7 fingers to get to the next number (21 + 7 = 28, then 28 + 7 = 35,  then 35 + 7 = 42, etc.).

With all of these strategies, students can get the correct answer, but they are often not really even using multiplication. Their method is often counting the objects in each group one at a time.  And when skip counting, if just one number is missed in the sequence then the total is obviously off. In addition, students often spend so much time with each of these that they get frustrated and give up.

In previous posts, I mentioned different ways for students to skip count while focusing on the patterns numbers make (Click HERE) and ways to use arrays to break it down into smaller equal groups (Click HERE).  So those methods are a little more productive toward using multiplication than the above. Today, though, I will steer you toward a unique strategy which does the following:

  • Allows students to use readily known facts (especially the 5s and 2s)
  • Adds a pictorial component which builds on subitizing, number sense, and decomposing of numbers
  • Applies the distributive property so students are using multiplication and addition together

The strategy modeled here is based on facts students already know. This is likely to be different among your students. Some will say they are great with their 4s or 3s. But most students I work with are proficient with their 5s and 2s (and can skip count quickly and accurately if they haven’t memorized these). So a lot of the problems shown will focus on use of 5s and/or 2s.

Again, let’s look at 6 x 7.  The student doesn’t know their 6’s and doesn’t know their 7’s. So we will decompose 6 or 7 to include a group of 5’s, which is known (I’ll show both ways).

  1. Decompose 6:  Six is made up of a group of 5 and a group of 1.  This is a pictorial method to build on subitizing using a dot pattern to show 5 and 1 (similar to a domino piece).
  2. See how the connection to the familiar ten frame can illustrate 7 x 6 (7 groups of 6) in this manner.
  3. Condense this concept to this representation which shows 7 x 5 plus 7 x 1 (35 + 7 = 42)

To see 7 decomposed instead of 6: Seven is made up of a group of 5 and a group of 2.

  1. See what this looks like on a ten frame to illustrate 6 x 7 (6 groups of 7):
  2. Condense to the “domino piece.” This shows 6 x 5 plus 6 x 2 (30 + 12 = 42):

Click on this link Multiplication Strategy pictorial CE for a FREE copy of the pictures above and below which are used in this post (for easy reference later). Here are a few more examples. Some use 5s and 2s, while others will show other combinations using 3s or 4s. The use of dots instead of numbers inside the “domino” is suggested to keep it a little more pictorial and less abstract. Plus, it builds on knowledge of subitizing (which is recognizing quantity without physically counting). Numbers alone can certainly be used, but the quantity of numbers might frustrate some students.

 

Practice activity:

  • Use a set of dominoes and digit cards 1-9. Turn over 1 domino and 1 digit card. Write the problem and then the decomposed version. See photo for example. Click on this link Digit cards 0-9 for a FREE copy of the digit cards.I’d love to hear if you are able to try this with your students. Let me know if it helps. I have worked with a couple of classes so far with this and they have loved it.  It opened a lot of eyes!!

Have a great week!

C. Elkins, OK Math and Reading Lady

Ten Frames Part 4: Multiplication

by C. Elkins, OK Math and Reading Lady

Yes, you can even use ten frames to teach multiplication concepts! Here are my mini ten-frames with dot cards from 1 – 10:  Click HERE to get a free copy. These are helpful to use, especially if you don’t have enough tens/ones blocks . . .  or you prefer manipulatives that are slightly easier to manage. These provide a strong connection to place value, and the commutiative and distributive properties.

I recommend two sets of the cards 1-9 per student. Each set has multiple copies of the same number. They can be laminated, cut, and placed in a baggie for ease in handing out and storage.

Multiplication Examples:

  1. Single digits (basic facts): 
    • For the problem 3 x 6, the ten frame is really helpful for the student to see 3 x 6 is almost like 3 x 5 with one more group of 3 added on (by being familiar with the fact that the top row on a ten frame is 5).
    • Because of the commutative property, I know these two facts will have the same answer. But which of these below do you think might be “easier” to solve? Students don’t often know they have a choice in how they can use the numbers to their advantage!
  2. Double digit x 1 digit:
    • Use of these also provides a strong connection of place value and multiplication. Notice how students can see the breakdown on the 4 x 12 problem (4 groups of 12 = 4 x 10 plus 4 x 2). Great introduction to the distributive property of multiplication!
    • Here is where application of the commutative property also comes in handy. Which of the methods below would you rather use to solve: count by 4’s or count by 12’s? Again, show students how to use their strengths to decide which way to think about solving the problem.
    • Even though the number of total pieces might seem to be a little overwhelming, it definitely is worth the effort for a few lessons so students get a visual picture of the magnitude of the products.
  3. Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.

Continue reading

Ten Frames Part 3: More addition, subtraction, and place value

by C. Elkins, OK Math and Reading Lady

Welcome back to Part 3 of my Ten Frame series. This will continue with some more ideas on using ten frames for addition and place value. Be sure to grab my free set of mini ten frame dot cards and Place value mat with ten frames to use with these activities.

Add 9:

How often do you see students counting their fingers, drawing tally marks, or other figures to add 9? But what if they could visualize and conceptualize adding 9 is almost like adding ten, but one less? This is where the ten frame comes in handy.

  • To be most efficient with adding 9, help students to add 10 (or a multiple of 10) to any single digit.  Example: 10 + 7, 20 + 4, 50 + 8 . . .
  • Show a problem such as 9 + 7 as part of your daily Number Talk. Observe and listen to how students are solving.
  • Introduce this strategy by showing two ten frames – one with 7 and the other with 9. Check for quick recognition (subitizing) of these amounts on each ten frame.
  • Move one counter from the ten frame with 7 to the ten frame with 9. This will complete it to a full ten frame. Then add 10 + 6 mentally.
  • The purpose is for students to visualize that 9 is just one away from 10 and can be a more efficient strategy than using fingers or tally marks.
  • Practice with several more +9 problems.
  • For 3rd and up try mental math problems such as 25 + 9 or 63 + 9.  Then how about problems like 54 + 19 (add 20 and take away one)?
  • Can students now explain this strategy verbally?

Subtract 9:

  • Let’s say you had the problem 14 -9.  Show 2 ten frames, one with 10 and one with 4 to show 14.
  • To subtract 9, focus on the full ten frame and show that removing 9 means almost all of them. Just 1 is left. I have illustrated this by using 2 color counters and turning the 9 over to a different color.
  • Combine the 1 that is left with the 4 on the second ten frame to get the answer of 5.
  • Looking at the number 14, I am moving the 1 left over to the one’s place (4 + 1 = 5). Therefore 14 – 9 = 5

Use of the ten frame provides a concrete method (moving counters around) and then easily moves to a pictorial method (pictures of dot cards). These experiences allow students to better process the abstract (numbers only) problems they will encounter.

Place Value Concepts: Continue reading

Ten Frames Part 2: Addition and subtraction

by C. Elkins, OK Math and Reading Lady

Last week’s focus was on using ten frames to help with students’ number sense and conceptual development of number bonds for amounts 1-10. This post will feature ways to use ten frames to enhance students’ understanding of addition and subtraction. Look for freebies and a video!

There are many addition and subtraction strategies to help students memorize the basic facts such as these below. The ten frame is a very good tool for students of all grade levels to make these strategies more concrete and visual. I will focus on some of these today.

  • add or take away 1 (or 2)
  • doubles, near doubles
  • facts of 10
  • make a ten
  • add or sub. 10
  • add or sub. 9
  • add or sub. tens and ones

Doubles and near doubles (doubles +1, -1, +2, or -2): If the doubles are memorized, then problems near doubles can be solved strategically. 

  • Show a doubles fact on a single ten frame (for up to 5 + 5).  Use a double ten-frame template for 6 + 6 and beyond.
  • With the same doubles fact showing, show a near doubles problem.  This should help students see that the answer is just one or two more or less.
  • Repeat with other examples.
  • Help student identify what a doubles + 1 more (or less) problem looks like. They often have a misconception there should be a 1 in the problem. Make sure they can explain where the “1” does come from. Examples:  7 + 8, 10+11, 24+25, 15 +16, etc.
  • For subtraction, start with the doubles problem showing and turn over the 2-color counters or remove them.

Facts of 10: These are important to grasp for higher level addition / subtraction problems as well as rounding concepts. Continue reading

Ten Frames Part 1: Number Sense

by C. Elkins, OK Math and Reading Lady

The focus in this post will be an introduction to ten frames and ways they can help your students gain number sense. Then stay tuned because ten frames can also be a great tool for addition, subtraction, multiplication, and division.

Subitizing: This is the ability to recognize an amount without physically counting. Looking at the picture of red counters: If the top row is full, does the student automatically know there are 5? Doing a Number Talk is a great way to practice subitizing using a ten frame:

  • Use your own or pre-made dot cards. Flash the card for 1-2 seconds. Observe students. Are any of them trying to point and count? Or do they seem to know right away? Here’s a great video I recommend: KG Number Talk with ten frames
  • Tell students to put their thumb in front of their chest (quietly) to signal they know how many there are.
  • Ask a few students to name the amount.
  • Then ask this very important question, “How did you know?”
  • For the top picture you might hope a child says, “I knew there were 5 because when the top row is full, there are 5.”
  • For the bottom picture, you might hope for these types of responses: “I saw 4 (making a square) and 1 more.” or “I saw 3 and 2 more.” or “I pictured the 2 at the bottom moving up to the top row and filling it up, which is 5.”

The idea is to keep building on this.

  • What if I showed 4 in the top row? Can the student rationalize that it was almost 5? Do they see 2 and 2?
  • What if I showed 5 in the top row and 1 in the bottom row? Can the student think “5 and 1 more is 6?”

Here are some resources you might like to help with subitizing using ten frames.

Number Bonds: Using ten frames to illustrate number bonds assists students with composing and decomposing numbers. Students then see that a number can be more than a counted amount or a digit on a jersey or phone number. Here is an example of number bonds for 6:

  • 6 is 5 and 1 (or 1 and 5).
  • 6 is 4 and 2 (or 2 and 4).
  • 6 is 6 and 0 (or 0 and 6).
  • 6 is 3 and 3.

Teaching strategies for number bonds using ten frames: Continue reading

Beginning of School Tips

by C. Elkins, OK Math and Reading Lady

I’m going to repost a few of my favorite beginning of the year articles along with some math and parent involvement tips (since last week focused more on literacy tips). I know this is coming to you on a Tuesday again this time (which is different than the normal Sunday release), due to some out of state travels (to see our grandson). I’ll get back on track here very soon.

  1. Here is a link to a post I made previously regarding a great back-to-school math/ literature activity:  Name Graphs with “Chrysanthemum” by Kevin Henkes

  2. Looking for some good stories to read to encourage classroom community (Grades K-5)? Try this post: Back to school stories and activities

I am in the middle of a great book study:  Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement by Steven Leinwand (Heinemann Publishers).  Click HERE  to get more details about the book. I’ll give you a rundown of what I’ve loved from this book so far:

  • The quality of instruction has more impact on student achievement than the curriculum or resources we use. This means the instruction is “enhancing, empowering, energizing, and engaging.”
  • “We can demonstrate, tell, and let our students practice, or we can engage and focus on understanding and application.”
  • Where do you fit? Where would you like to be? Which model provides students with the opportunity for productive struggle?
    • The more traditional:  Teacher instructs, teacher solves example problem with class, students practice on their own while teacher assists those who need help.  Or . . .
    • The focus on understanding: Teacher poses a problem (though-provoking). Students struggle. Students present ideas to class. Class discusses various solutions. Teacher summarizes class conclusions. Students practice similar problems.
  • Teacher questions like “Why?” and “How do you know?” invite students to explain their thinking and show different ways to solve a problem.
  • Daily cumulative review is important.  (I will touch more on this in later posts on ways you can incorporate this into your daily math routine where it is interesting, informative, and engaging. In the meantime, check out the categories section of my blog “Number Talks and Math Meetings“).

Miscellaneous parent involvement tips:

One of my goals the year I worked on National Board Certification was to improve parent involvement. In the last post I mentioned keeping a log of parent contacts and writing a weekly or monthly class newsletter or blog. Here are two other things I initiated that proved very successful, so I thought I’d share them with you.

  1. Invite parents to write to you about their child.  At the beginning of the year, I asked parents to write a note telling me about their child. I invited them to tell me the special things they wanted me as the teacher to know – to include their successes and proud moments. Perhaps even share the goals they have for their child, information about siblings, their feelings about homework, etc. This information was helpful to me to get to know the child better. Parents really appreciated the chance to tell about their child, and it set the stage for open communications with the parents. I hope you will try it.
  2. With the students’ help, we put together a memory book of the year’s events at school. I took lots of pictures (even of routine things like eating lunch, lining up, library time, where we put our coats, etc.). Every couple of months I printed the pictures and students chose 1 or 2 to write about. After editing the writing, the pictures and written captions were put together in a memory book (big scrapbook). We added borders, stickers, and other scrapbooking type visuals. We tried to finish the main parts of it by February so it was ready to share with the parents. It was available for viewing at conference times, and students could check it out to take home for parents to see.  It was especially valuable to those parents who were not able to visit school.  I put a few comment pages in the back for parents to leave notes. You wouldn’t believe how many had a much better understanding of the complex day-to-day school events and appreciated the chance to see what really goes on at school all day. After 2-3 years of making a book version,  I changed it to a digital format (power point) instead of a book version (because parents wanted copies). With a digital version, you have the capability of importing graphics, etc. to make it “fancy.” I still have my books and will always cherish them.

Enjoy!! Coming soon — I’ll share more from the book “Accessible Mathematics” as well as some cool things I’ve learned from a Building Math Minds summit I attended.

Be sure invite some of your new teachers to join this blog.

 

 

 

Multiplication Concepts Part 5: Multiple digit strategies

by C. Elkins, OK Math and Reading Lady

In this post, I will share some strategies for using concrete manipulatives and pictorial methods to solve multiple digit multiplication problems. By using these methods, students gain a better sense of place value as they work to decompose the problem into smaller units.  Decomposing also allows a student to better perform mental calculations. Some helpful manipulatives:  base ten materials (hundreds, tens, ones); place value disks; cups and pinto beans

What is the purpose of knowing multiple strategies? Some would argue that too many strategies are confusing for students. Some believe the only strategy needed is the standard algorithm. I believe teaching different strategies provides students with choices and improves analytical thinking. With only 1 strategy, if the “steps” are missed, the student has no other recourse. Student choice is a powerful motivator as well because they get a say-so in how they approach their own work.

I keep thinking about my past teaching when I only taught the standard algorithm (before I knew better). I recall saying: “Show all your work – because I said so.” This means I was not considering the students who were able to do some of the mental calculations in their head. I know I went through the steps in a robotic, don’t-question-me way:  “Multiply the ones, carry to the ten’s place, multiply again and add the digit you carried. When multiplying the 2nd digit, be sure to watch the placement in the second row and scoot it over to the left one place.” None of this conversation (if you could even call it that) mentioned the place value relationship, what the carried digit represented, or why the second row of the answer should be scooted over one place.

Here are some examples relating manipulative and pictorial methods with paper-pencil methods. I’ll use the problem 32 x 4. These methods help students use (30 + 2) x 4 to solve.

  1. Base ten: Show 3 tens rods and 2 ones four times.
  2. Place value disks: Show three 10’s disks and two 1’s disks four times.
  3. Cups and beans: Each cup contains 10 beans. Ones are shown by individual beans. Show 3 cups and 2 beans four times.
  4. Pictorial drawings and decomposing models:
  5. Partial products: This is a great way to help student realize that the 3 represents 30.
  6. Area (box) model: Another ways to visualize and utilize place value knowledge to solve.

When it is time to introduce the standard algorithm, you can relate it to the partial products or area model. I always recommend showing both side by side so students now understand what the carried digit represents, and why the second row is scooted over to the left, etc. Then try some problems like this for your daily mental math number talks (show problem horizontally). I practically guarantee that students who can visualize the manipulatives or the partial products method will get the answer more quickly than those who are performing the std. algorithm “in the air.”

I will take a break this summer and come back every now and then between now and August. Keep in touch! Enjoy your summer!!! Let me know if there are topics you’d like me to address on this blog.

Multiplication Concepts Part 4: Skip Counting

by C. Elkins, OK Math and Reading Lady

This is part 4 in a continuing series of posts about basic multiplication teaching concepts. Use them for beginning lessons or reteaching for struggling learners. Students could be struggling because they were not given enough exposure to concrete and pictorial models before going to the numbers only practices. The focus in this post will be skip counting to determine multiplication products. I will even focus on skip counting done in early grades (counting by 10’s, 5’s, and 2’s). Read on for 10 teaching strategies regarding skip counting.

I am going to give some of my opinions and misconceptions students have about skip counting.

  • Many students do not associate skip counting with multiplication, but just an exercise they started learning in KG and 1st (skip counting orally by 10’s, 5’s, and 2’s).  This is often because they started with numbers only and did not have the chance to see what this looks like using concrete objects or pictorial representations.
  • If you observe students skip counting, are they really just counting by 1’s over and over again? Or are they adding the number they are skip counting by repeatedly.  You know the scenario. You tell a student to skip count by 3’s and they know 3, 6, 9, but then hold up their 3 fingers and count 10, 11, 12, 13, 14, 15, 16, 17, 18, and so on.  Or are they truly counting like this: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30?
  • The main issue I have with skip counting is that if a student makes an error regarding just one of the numbers in the sequence, then the rest of the sequence is incorrect. So this should not be their only strategy. Do you recall a previous story I mentioned about the 5th grader who tried to solve 12 x 3 by skip counting on a timed facts test? He was unsuccessful because he kept losing track and didn’t have another strategy to use.
  • Successful skip counting reinforces the concept that multiplication is repeated addition – do your students know this? I have witnessed many students who know the first 2-3 numbers in a skip counting sequence, but then don’t know how to get to the next numbers in the sequence.
  • Students don’t often relate the commutative property to skip counting. Let’s say the problem is 5 x 8. The student tries skip counting by 8’s (because this problem means 5 groups of 8) and may have difficulty.  Does the student try to skip count by 5’s eight times instead?

Ten teaching strategies for skip counting:

  1. For young students skip counting, use objects to show how to keep track:
    • Base 10 rods
    • Rekenrek (easily slide 5 or 10 beads at a time)
    • Hand prints (for counting 5’s or 10’s):  Which do you think would give students a better understanding: Holding up one hand at a time and counting by 5’s or lining up several children and having them hold up their hands as you continue counting? The second scenario enables students to see the total of fingers as opposed to just 5 at a time.
    • Use money: nickels and dimes
    • Associate counting by 2’s with concepts of even and odd
  2. Use manipulatives.  Do it often and with a variety of materials. The arrangements should emphasize the other strategies (equal groups, arrays, repeated addition).
  3. Draw and label pictures. The labels for this strategy would show the cumulative totals instead of just the number in each group.
  4. Arrange students in line or groups to practice skip counting. Example if practicing 4’s: Every 4th student turns sideways, every 4th student holds up their hands, every 4th student sits down. every 4th student holds a card with the number representing their value in the counting sequence, etc.
  5. Practice skip counting while bouncing or dribbling a ball. Great for PE class!
  6. Associate skip counting with sports:  2 and 3 pointers in basketball, 6 points for touchdowns in football, etc.
  7. Use a 0-100 chart to see patterns made when skip counting. I love the 0-100 pocket chart and translucent inserts that allow you to model this whole group. Individual 100 charts are readily available in which students can mark or color the spaces. Here are links to the chart and the translucent inserts: 1-100 pocket chart and Translucent pocket chart inserts

     

  8. Look for other patterns regarding skip counting. Refer to my previous post on this for more details: Skip counting patterns

     

  9. Relate skip counting to function charts and algebraic patterns using growing patterns.
  10. Practice skip counting using money: by 5’s, 10’s, 25’s, 50’s

What strategies do you like for multiplication? What misconceptions do you see with your students?

Next post will be part 5 of my multiplication posts – and the last one for this school year. I will focus on using these basic concepts with double-digit problems. Stay tuned!!

 

Multiplication Concepts Part 3: Equal Groups

by C. Elkins, OK Math and Reading Lady

Thanks for checking in on part 3 of my multiplication posts. Focus will be on the equal groups strategy — looking at how students can efficiently use this strategy to help learn basic multiplication facts. My angle will be at the conceptual level by using concrete and pictorial methods.

Basics:

  • Instead of in array or area format, equal groups are separate groups.
  • The “x” means “groups of.”  So 3 x 4 means “3 groups of 4.”

What things normally come in equal groups? Conduct a brainstorming session. I love the book “What Comes in 2’s, 3’s, and 4’s” as a springboard. After reading the book, let students brainstorm other things that come in equal groups. See the pictures below for some more ideas. After some internet research, I also made this attached list to use (in case you or your students draw a blank): click here: Equal groups pictures and list template

Use these lists to help students generate stories about equal groups. When students can create (and maybe illustrate) their own stories, they are much better at solving problems they must read on their own. This also helps students think carefully about what in the story constitutes a “group” and what the “groups of” represents:  

  1. There were 5 bowling balls on the rack. If you count all of the holes (3 per ball), how many holes are there all together? (5 x 3). The bowling balls are the groups. The holes are what is being counted in each group.
  2. How many numbers are shown on 3 clocks? (3 x 12). The clocks are the groups. The numbers are what is being counted in each group.
  3. I bought 8 pair of earrings. How many earrings are there? (8 x 2). The pairs are the groups.
  4. Seven ladybugs were crawling on the leaves. How many legs would there be? (7 x 6). The ladybugs are the groups. The legs are what is being counted in each group.

Ways to show equal groups with objects and drawings:

  • Hula hoops (great to use these in PE class to emphasize multiplication)
  • Embroidery hoops
  • Circles of yarn
  • Dishes:  cup, bowl, plate, tray
  • Baskets
  • Shelves

Objects to use to show equal groups:

  • people
  • cubes
  • tiles
  • mini erasers
  • teddy bear manipulatives
  • base ten materials
  • food: pinto beans, macaroni, cereal, candy
  • practically anything you have an abundance of!!

Teaching concepts regarding equal groups:

  • When students are placing objects or drawing inside, do they randomly place objects? Or do they organize them to enable ease in counting? Showing students how to organize the objects in each set contributes to their knowledge of equal groups — AND it’s a big help to you as the teacher as you check on students. If the dots are randomly placed, the teacher and student must count one at a time to check. If they are organized, teacher and student can tell at a glance if the amount in each group is correct. Notice the difference below: Which ones show a student’s understanding of 9? Which ones can a student or teacher check rapidly?

  • When counting the objects or drawings to determine the product of these equal groups, are students counting one at a time? Or are they counting in equal groups (such as by 2’s, 5’s, 3’s, etc.)? If we allow students to just count by ones, then they are not practicing multiplication, just counting!!

Activities to practice equal groups strategy:

  1. Circles and Stars:  Roll a dice once. This is the number of circles to draw. Roll a dice again. This is the number of stars to draw inside. If played with a partner, students can keep track of their totals to determine a winner. Dice can be varied depending on the facts that need to be practiced. A spinner can also be used. (See picture at beginning of this post.)
  2. Variation of above:  Use other materials (such as those listed above).
    • Dice roll #1 = # of cups. Dice roll #2 = number of cubes
    • Dice roll #1 = # of hoops. Dice roll #2 = # of pinto beans
    • Dice roll #1 = # of plates. Dice roll #2 = # of Cheerios
  3. Write and illustrate stories:  Provide a problem for students to illustrate (example:  6 x 3 or 3 x 6).  Then each student can decide how to form the story and illustrate. I always tell students to choose items they like to draw to make their story. Here are some examples.  See some examples from former students.
    • There were 6 monsters in the cave.  Each monster had 3 eyeballs. How many eyeballs all together?
    • Six princesses lived in the castle. They each had 3 ponies. How many ponies in all?
    • There are 3 plants in the garden. They each have 6 flowers. How many flowers are in my garden?
    • I made 3 pizzas. Each pizza had 6 slices. How many slices of pizza did I make?
  4. PE Class activities:  If your PE teacher likes to help you with your learning objectives, let them know you are working on equal groups strategies. While I’ve not done this personally, I think having relay races related to this would work perfectly. For example, the teacher presents a problem and each team must use hula hoops and objects to show the problem (and the answer).
  5. Try these story books about multiplication:
  6. Equal groups story problems to solve:  See my previous post related to this. You will find some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click HERE

Enjoy!!  

 

Multiplication Concepts Part 1: Repeated Addition

by C. Elkins, OK Math and Reading Lady

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

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

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

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

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

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

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

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

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

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

Geometry Websites

by C. Elkins, OK Math and Reading Lady

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

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

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

What are the possibilities with this?

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

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

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

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

4. “Lines” on GeoGebra

5. “Angles” on GeoGebra

6. “Plane Figures” on GeoGebra

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

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

Graphic Organizers for Math

by C. Elkins, OK Math and Reading Lady

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

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

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

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

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

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

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

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

Division Basics Part 3: Repeated Subtraction and # Line

by C. Elkins, OK Math and Reading Lady÷

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

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

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

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

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

Division Basics Part 2: Equal Groups

by C. Elkins, OK Math and Reading Lady

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

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

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

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

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

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

 

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

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

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

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

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

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

Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady

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

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

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

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

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

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

Continue reading

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

by C. Elkins, OK Math and Reading Lady

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

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

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

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

Ideas:

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

Continue reading

Geometry Part 7: Area and Perimeter

by C. Elkins, OK Math and Reading Lady

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

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

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

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

Ideas:

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

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

Continue reading