Multiplication using Ten Frames or Base Ten

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.

After students get practice using these manipulatives (concrete), then proceed to pictorial models to draw them as simply as possible.  This will give them a good foundation to apply to the abstract (numbers only) problems.  I always pitch for the CPA progression whenever possible!!!

I will pause a while for the summer and just post once a month until school starts up again.  Take care, everyone!  But please don’t be shy.  Post your comments, ask your questions, etc.

Multiplication strategies — Equal groups

by C. Elkins, OK Math and Reading Lady

Thanks for checking in on another multiplication strategy! The focus for this post 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. Be sure to see the links at the end for books and my free equal groups story cards.

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:  Here are some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click to see the blog post on equal groups story problems and get my FREE set of story problem cards:  HERE

Enjoy!!  Many of you are now off for a well-deserved summer break. Use your summer time to catch up on my postings from this past year, or email me for more information. Also, feel free to comment on any article about your experience or additional tips. 

Multiplication strategies — using arrays

by C. Elkins, OK Math and Reading Lady

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

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

Array Basics:

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

Ways to incorporate arrays into story problems:

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

Activities to encourage concrete and pictorial construction of arrays:

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

     

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

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

Multiplication: Repeated addtion

by C. Elkins, OK Math and Reading Lady

The next few posts will continue to 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, decomposing, and doubling/halving models as well as the associative and distributive properties.  Freebies below!!

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. . . and encourage counting in equal groups to promote a growth of the multiplication mindset.

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 (using his fingers). 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.

Here are a few resources (FREE) that might help with this strategy:

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”).

Multiplication — Developing an understanding

by C. Elkins, OK Math and Reading lady

Are you looking for some ways to help your students learn the multiplication facts? Or ways to help them solve multiplication problems while they are in the process of learning the facts? One way is to skip count or repeatedly add the number over and over again. While this is one acceptable strategy, I see many students skip count using their fingers, often starting over numerous times. And if the child miscounts just one number in the sequence, then all of the remaining multiples/products are incorrect. Sometimes the student will write down the sequence in a horizontal row (better than using fingers in my opinion), but again – if they miss one number . . . all the rest of the numbers in their list are wrong.

What I want to show you today (in Part I of my series about multiplication strategies) are ways to relate the multiples/products in recognizable patterns which may facilitate recall and help with committing the facts to memory. Yes, students should also know the following about multiplication – and I will focus on all of these in future articles:

  1. Multiplication is repeated addition. For example: 3 x 4 means 3 groups of 4 or 4 + 4 + 4 = 12
  2. Multiplication is equal groups. 3 x 4 might be shown with 3 circles and 4 dots in each one. Be cautious about continued use of this one. Students are good at drawing this out, but then are they actually adding repeated groups or just counting one dot at a time. Observe students to see what they are doing. Transition to showing 3 circles with the number 4 in each one.
  3. Multiplication is commutative. If solving 7 x 2 (7 groups of 2), does the student count by 2’s seven times, or perhaps make it more efficient by changing it around to make it 2 x 7 (2 groups of 7 — and adding 7 + 7)?
  4. Multiplication can be shown with arrays. If students are drawing arrays to help solve, watch how they are computing the product. Are they counting one dot at a time? Or are they grouping some rows or columns together to make this method more efficient. I will focus on this one in Part 2.
  5. Multiplication facts have interesting relationships — stay tuned for future posts or check out my blog archives for more
    • An even number x an even number = an even number
    • An odd number x an even number = an even number
    • An odd number x an odd number = an odd number
    • 2’s, 4’s, and 8’s are related
    • 5’s and 10’s are related
    • 3’s, 6’s, and 9’s are related
  6. Multiplication can be shown by skip counting.  Aside from my comments above about errors students make with skip counting, there are some ways to arrange skip counted numbers in distinctive organized groupings so the patterns become more noticeable, perhaps leading more to memorization than just a horizontal list of numbers, or using fingers.
    • I have included 2 visuals to see some of my favorite ways to relate skip counting to unique patterns (for the 2’s, 3’s, 4’s, 6’s, 8’s, and 9’s). Visualizing and explaining the patterns is a good exercise for the brain.  Can your students come up with another way to visualize the patterns with these numbers?

 

Stay tuned for more blog entries about multiplication!

Interactive math lessons and activities on NCTM

Review by C. Elkins, OK Math and Reading Lady

Resource – http://illuminations.nctm.org

This is a math resource I absolutely love! It is a product of the National Council for the Teachers of Mathematics (NCTM).

This site includes lesson plans and interactive activities. Search in several ways: by topic, by standard, or by grade level. Need some strategy games? Check out “Calculation Nation” (some of which can be played against other players), and “Brain Teasers.” I have just added this link to my Resources page (on my blog home page).  Pass this along to parents for them to use with their children at home!

Many of the lessons connect to exploration projects and literature. The interactive features are outstanding!! These are perfect for the smartboard, on laptops, or tablets. Doing Zoom lessons? Then these are also wonderful for sharing the screen to introduce or review concepts. Once you are on the home screen, click the Interactives box (right side) and then the desired grade level. There are dozens of great applets, but here are a few you might really like. I have linked them for easy reference, so just click on the  title and you’ll be there:

Dynamic Paper: Customize graph paper, number lines, spinners, nets, number grids, shapes (to include pattern blocks, color tiles, and attribute blocks), and tessellations. You can also choose inches or cm. These can be customized, saved and printed as jpeg or pdf. I created the spinner shown here from this application.

Five Frame and Ten Frame tools: Geat activities to build number sense using five or ten frames. These may take 1-2 minutes to load.

Cubes: Build a rectangular prism one cube, or row, or layer at a time and then compute the volume or surface area.

Coin Box: Drag and exchange coins. There is also a feature I like (the grid at the bottom right corner), which puts coins in blocks (by 1s for pennies, 5s for nickels, 10s for dimes, and 25s for quarters). This really helps see the value of the coins. Want more info about coin blocks? Once on the Coin Box page, click on the “Related Resources” tab.

Try these for fractions: Fraction Models (which includes decimal and percent equivalencies) and the Fraction Game.

Geometric Solids: Create a shape (either transparent or solid) and swivel it around to see all of the faces, vertices, and edges. It has a cube and pyramid as far as basic 3D shapes are concerned. I wish it had more that 3rd-5th students would encounter.

Here’s a nice multiplication game:  Product Game  Two players (or a player vs. the computer) choose factors from the bottom bar to create products shown on the game board grid to get 4 in a row (and try to block your opponent from getting 4 in a row).  Be sure to see the directions included.

Some of the interactives require an NCTM subscription.  The ones I have listed above should be okay to access without membership. I have subscribed for years and just paid $94 for this coming year. Well worth it if you plan on using their site extensively.  This subscription also entitles you to a print and and online journal, blog capabilities, and more.

Enjoy these and so many more!!! Let us know if there are others you recommend.  I’ll highlight more on my next post.

 

Rounding activities (whole numbers and decimals)

by C. Elkins, OK Math and Reading Lady

Last week I reposted my blog regarding use of number lines to assist students with number sense and rounding. Check it out for free activities and rounding charts. Today I am sharing some more rounding activities I developed and used with students to practice (with either whole numbers or decimals). These activities can be varied to suit your students’ needs.

These grid templates are to use the activities with 2-4 students (or teacher vs. student if working one-on-one online). I developed 3 different grid sizes (4 x 4, 5 x 5, and 6 x 6).  You will also need something to generate numbers for each set of players:

  • Grid for playing board:  Get here FREE  Grid 4 x 4   Grid 5 x 5   Grid 6 x 6
  • 2 dice (1-6)
  •  2 dice (1-9)
  • digit cards (0-9) — get your free set here:  0-9 digit cards
  • deck of playing cards (with tens and face cards removed)
  • spinner (with digits 0-9) — 1 is ok, 2 is better

The objective of the game is for a player to capture 4, 5, or 6 squares in a row (horizontally, vertically, diagonally).  You decide based on the size of the grid and the skill level of the players how many captured squares are needed.

The teacher can write in possible answers on the grid and laminate for continued use (samples below). Then students can use a game piece  (flat stones, two-color counters, etc.) or different color dry erase marker to mark their square.

  • Using a paper form, students can write in answer choices randomly on the grid (supplied by the teacher for accuracy). Then each player can use a different colored crayon to mark their square.

Here are some different variations of the game (whole number rounding to nearest 10, 100, 1000 and decimal rounding to the nearest tenth or hundredth).

Rounding to the nearest ten:  You can use the blank grid to write in your own numbers randomly.  Consider which number generated options you are using.  If you use 1-6 dice, the biggest number on the board has to be 70 and remember there’s only 2 ways to achieve 70 (by rolling a 6 and 5 or a 6 and 6).  If you use 1-9 dice or number cards, then you can place numbers from 10-100 on the board.  This gives a few more options and a chance to round higher numbers.

  • Roll 2 dice (or turn over 2 number cards, spin spinner twice)
  • Generate a 2 digit number.  If a 3 and 5 are rolled, the player can decide to make it 35 or 53.
  • Round that number to nearest 10.
  • Find that number on the grid.
  • If using a laminated board, place a colored “chip” on it. If using paper, each player colors their chosen # with a crayon.
  • Player #2 follows same steps.
  • Each player is trying to get 4, 5, or 6 in a row (depending on which grid size you choose).
  • It’s more fun if you try to block the other player and use strategies about your choice of a number to round (should I use 35 — rounded to 40?  Or 53 — rounded to 50?)

Rounding to the nearest hundred:

  • Follow same steps as above, except use 3 dice or 3 number cards.
  • Place numbers such as 0, 100, 200, 300 . . . randomly on the board. In the samples pictured I numbered to 1000 since I used 0-9 dice. I didn’t show a 0 on the boards pictured below, but should have since a number less than 50 could actually be generated. If using 1-6 numbered dice, the highest would be 700.
  • Example:  Roll a 2, 5, 6 — player can make these numbers 256, 265, 526, 562, 625, 652.  The number choice becomes part of the strategy of the game to see which spot is available on the board.

Rounding to the nearest thousand:

  • Follow same steps as above, except use 4 dice or 4 number cards.  If using 1-6 numbered dice, the highest would be 7000.

Rounding to the nearest tenth:

  • Follow steps similar to rounding to nearest tenth, except answer choices on the grid would look like this:  .1, .2, .3 . . .
  • If using number cards (as pictured below) or a spinner with digits up to 9, be sure to include a space on the grid for 1 (which is what you would round these numbers to:  .95, .96, .97, .98, or .99.
  • Again, be mindful of randomly placing numbers because it depends on which number generating options you are using.  If using 1-6 dice, I would only include a couple of spaces with .7 because there’s a limited number of ways to round to .7 with dice numbered 1-6.  The only way to round to .7 would be to roll a .65 or a .66.

Rounding to the nearest hundredth:

  • Follow steps similar to rounding to the nearest hundred by using 3 dice or turning over 3 number cards.  Be sure to include a space or two for an answer of 1.

Other tips for playing:

  1. Provide students with a blank white board to draw an open number line to check out their answer.
  2. Provide a sentence frame such as:  I made the number  ______ which is rounded to ________.
  3. Remind the players that it is their job to watch their opponent and challenge anything they think may not be correct (in a friendly, helping manner of course).
  4. Shorter time frame for playing?  Choose the 4 x 4 grid.  Longer time frame?  Choose the 6 x 6 grid or use the 6 x 6 grid with the winner being one to get 5 in a row.
  5. Consider creating a box of 4 completed squares in addition to 4 in a row.
  6. This can be played as teacher vs. students in a virtual setting.
  7. This can be played in a one-on-one online setting by using a document camera or posting a screen shot on the screen.

Let me know if you try these!  Pass along any extra tips you have.

Also, a reminder to contact me if you would like personalized professional development over any reading or math strategy.  I can do a Zoom session with you or a group of teachers.  Flexible payment options.  Also, check out my link on the side bar for Varsity Tutors regarding the opportunity for you to tutor students online or in person (and earn a bonus for using my name).

Take care, stay safe!!!  

 

Rounding and Number Lines

by C. Elkins, OK Math and Reading Lady 

I get requests from many teachers to help with instructional strategies regarding rounding, so I am happy to share my thoughts (and freebies) with you. Difficulty with rounding usually means students lack number sense. The essential goal of rounding is: Can you name a benchmark number (whole, tens, hundreds, thousands, tenths, hundredths, etc.) that a given number is closer to? I have found the more experience a student has with number lines, the better they will be with number sense, and the better they are with rounding to the nearest ___.  Then this rounding practice must be applied to real world problems to estimate sums, differences, products, or quotients.

When doing a google search for tips on rounding (ie Pinterest), you very often find an assortment of rhymes (such as “5 or more let it soar, 4 or less let it rest”) and graphics showing underlining of digits and arrows pointing to other digits. These steps are supposed to help children think about how to change (or round) a number to one with a zero. Many students can recite the rhyme, but then misunderstand the intent, often applying the steps to the wrong digit, showing they really don’t have number sense but are just trying to follow steps.

My answer (and that of other math specialists) is teaching students how to place any number on a number line, and then determining which benchmark number it is closest to. Continue reading to see examples and get some free activities. And watch next week for some new rounding activities for grades 2-6 (whole numbers and decimals).

Continue reading

Number Talks with Dot Cards: Subitizing, Number Sense, Facts (Part 2)

by C. Elkins, OK Math and Reading Lady

Hi!  This is Part 2 regarding ways to do number talks using dot cards. This post will feature random dot cards. See the last post for strategies with ten frame dot cards and some background information about why and how (click HERE).

My pictures below feature dot cards provided via an extra purchase from this great resource regarding Number Talks. I blacked out the number in the small print at the bottom of each card because I was using them online and didn’t want the magnification to show the number.  When showing them in person, the number is too small really for a student to notice or I can use my hand to cover it when showing the card.  Anyway . . . that’s for those of you wondering what the little black smudge was. Here’s an amazon link to the cards which you can get digitally for $19.95 (279 pages worth): Number Talk Dot Cards

My previous post (linked above) also listed 2 resources for ten frame and random dot cards.  Here is another one you might like and is great to use with partners as well.  I’ll describe an activity with them below.  Dot Cards for Number Sense ($2 from mathgeekmama.com)

You may like checking out mathgeekmama for other wonderful FREE resources.

Random Dot Cards

While I refer to these as “random” dot cards, it really doesn’t mean the dots are just scattered willy-nilly.  The dots on these cards are still organized, but just not on ten frames.  When using these cards, the goal is for students to “see” patterns with the dots to aid their subitizing and quick recall of number pairs.  You might start with dot dice first, then look for these on the dot cards:

  • groups of 2
  • groups of 3 (such as triangles)
  • groups of 4 (such as squares)
  • groups of 5 (like on a dice)
  • groups of 6 (like on a dice)
  • doubles
  • near doubles

I also often point out to students how I mentally “move” a dot to visualize one of the above scenarios. This will be shown in the pictures below with an arrow.

Procedures for whole group (either in person or on Zoom):

  1. Flash the card (longer for more dots).
  2. Students put thumb up (I prefer thumb in front of chest) when they have decided the amount.
  3. Randomly select students to tell you how many they saw. No judgement yet on who is correct and who isn’t.
  4. Then ask the VERY important question, “How did you see it?”  This should elicit various responses which will help reinforce different ways numbers can be decomposed.
  5. If desired with in-person sessions, you can have students pair-share their response first before calling on students to tell you. This way all students get a chance to share their way with a ready listener.  Click on this link for a way to silently signal  “Me too” in sign language. I find this very helpful especially for those students who want to respond — and helps avoid the “he took my answer” complaint.
  6. Record the different responses on a chart tablet.
  7. On the occasions where there are limited responses, here are some options:
    • Ask students if they see a way another student might have seen it. Be prepared — you might get some amazing (or long-winded) responses.
    • If students don’t see something I think it worth mentioning, I might say, “Here’s a way I saw a student think about this one last year.”
    • Or you could  just show the card another day to see if there are some new responses then.

What do you see with these?  . . . Plus some examples:

How do you see these? . . . Plus suggested outcomes:

Procedures for individual or partners (great for online tutoring or class center activity)

  1. Flash the card (longer if more complicated).
  2. Student tells you how many.  If not correct, show the card again.
  3. Ask, “How did you see them?”
  4. If the card is laminated, circle the parts the child describes.
  5. Tell how you (teacher) saw it.
  6. Ask, “How might another student see it?”  This gets them to see other possibilities.
  7. Record responses.

With the activity I mentioned earlier from mathgeekmama.com, this is a great with partners. I would recommend dot cards with no more than 8 dots for this activity:

  • Start with a stack of dot cards (face down).  Provide a blank laminated square to record dots on.
  • Partner 1 selects the top card and flashes it to partner 2 (perhaps 2-3 seconds).
  • Partner 2 uses a laminated blank square to try to draw the dots (with dry erase marker) to match what partner 1 showed them.
  • Both students reveal their dot cards to see if they match.
  • Switch roles and repeat.

As an individual activity, provide the laminated dot cards and a dry-erase marker.  Circle the dots.  Write a math problem to match it. Take pictures to record answers. (Recommendation: Do this after you have already modeled it during a Number Talk session.)

Take care. Share your experience with using dot cards for Number Talk sessions. I love success stories!

Interesed in personal professional development, or PD for your grade level team or school? Please contact me for special rates. I can meet via Zoom for just about any need you have (math or reading).  I’d love to help!

Number Talks with Dot Cards: Subitizing, Number Sense, Facts (Part 1)

by C. Elkins, OK Math & Reading Lady

Do you see 3 + 4 =7 or perhaps 5 + 2 = 7? Maybe you see 3 + 2 + 2 = 7.

I have been using dot cards for many years with K-2 students as part of my Number Talks routine. I’d like to share some ways to follow this routine using both ten frame dot cards and random dot cards.  These are also easy to use via distance learning situations.

If you haven’t tried this before, you are in for a treat!  It is so nice to listen how students process their thinking. I never cease to be amazed at how developed a child’s thoughts can be expressed . . . and how many children take this as a challenge to see how many ways a dot picture can be explained.  I often feel I learn so much about my students capabilities (or sometimes the deficits) during this type of Number Talk session.  Look for my recommended links below (FREE).

What are the benefits?:

  1. Students gain the ability to subitize (tell a quantity without physically counting).
  2. Students gain number sense by noticing more dots, less dots, patterns aid counting, the same quantity can be shown different ways, sequencing numbers, skip counting, and many more.
  3. Students gain the ability to see many different ways a number can be composed or decomposed which assists with addition and subtraction facts.
  4. Students gain practice with strategies such as counting on, add/subtract 1, doubles, near doubles, adding 9, adding 10, missing addends, and equal groups.
  5. Teachers are able to observe students’ processing skills in an informal math setting.

Materials needed:

  1. Ten frame dot cards:  This set is FREE from TPT and includes ten frame cards as well as random dot cards. Great find!!  https://www.teacherspayteachers.com/FreeDownload/Number-Talks-Early-Level-Starter-Pack-10-Frames-and-Dot-Cards-4448073
  2. Random dot cards (not on ten frames)

General procedures:

  1. Decide how you are going to show the cards:
    • Show to students who are seated near the teacher?
    • Show to students via a document camera projected to a screen?
    • Show to students online with a split screen?
    • Show to students via a ppt?
  2. Depending on the grade level, you may want to flash the card quickly to encourage subitizing or shorten/extend the time the card is shown.
    • To encourage subitzing to 5, I recommend flashing the card for a couple of seconds for dots from 1-5 for all age groups.
    • Depending on the number of dots and the complexity of the dots, you may choose to shorten or extend the time you display the card for amounts more than 5.  The goal is for the students to look for patterns, equal groups, doubles, dots making squares, rectangles, or triangles, determine a quantity, and then explain how they arrived at that amount.
  3. Students put a quiet thumbs up when they have decided the quantity.  They should not say the amount outloud at this point. This shows respect for others who are still processing.
  4. The teacher observes to see who is counting, who is participating, who uses fingers, who is quick /slow, etc.
  5. Teacher asks random students, “How many dots?”
  6. Teacher asks random students, “How did you see them?”
  7. Results can be stated verbally or written down by the teacher.

Here are some examples with sums less than 10:

Here are examples using 2 ten frames to illustrate quantities greater than 10:

Next post:  I will feature ways to use the random dot cards for your Number Talk sessions.

Do you need professional development for yourself, your team, your school?  Please contact me and we can work out a plan that fits your needs.  I can provide personal help via email or Zoom all the way up to custom made webinars or power point presentations.  Let me know!

Do you know students who need extra help at home via online tutoring?  See my link for Varsity Tutors and mention my name. 

Do you want to do some online tutoring yourself at a time that works with your schedule? See my link for Varsity Tutors and mention my name.  Feel free to ask me questions as well.  

 

 

 

Number Pairs / Number Bonds Activities (PreK-2): Part 2

by C. Elkins, OK Math and Reading Lady

This post will feature some more number pairs / number bonds activities as well as ideas for informal assessment (along with some FREEBIES).  See the previous post for Part 1.  Also, here is another cool virtual manipulatives site:  https://toytheater.com/category/teacher-tools/  You will find lots of materials for students to use to help with these activities:  counters, bears, two-color counters, whole-part-part templates, Rekenreks, etc.  Check it out!

For all of these activities, the student should be working with the number of manipulatives to match their focus number.  They should do several different activities using that same amount to get lots of different experiences making the same number pairs repeatedly.  After a generous amount of practice, assess the child and move to the next number when ready. An important feature of each activity is for the student to verbalize the combination being made. Using a sentence frame they can have with them or putting it on the board for all to see is a plus:  “____ and ____ makes _____.” Students will usually need reminders that you should hear them saying this.  It takes if from just playing to being cognizant this is a serious math activity.

  1. Heads or Tails:  Use coins and a whole-part-part template.  The student shakes and gently drops some coins (stick to one type of coin). Then sort according to how many landed on heads vs. tails by placing them on one of the templates.  Say the combination outloud:  “5 heads and 2 tails makes 7.”  Repeat.  Here’s a FREE Coin Toss recording sheet.
  2. Paper Cups:  The student finds different ways to place small paper cups up or down to match their focus number.  Example:  To make 7 I could have 5 up and 2 down, or 6 up and 1 down, or 4 up and 3 down, etc.
  3. Hiding or “Bear in the Cave”:
    • Use a small bowl, clean plastic butter tub, etc. and some objects (cubes, stones, beans, cheerios, M&Ms).
    • With a partner and the number of objects matching the student’s focus number, partner 1 closes their eyes while partner 2 hides some of the counters under the tub and the rest outside or on the tub.
    • Partner 1 opens his eyes and names how many outside the tub and then tries to determine the number hiding.
    • Partner 2 can then reveal if partner 1 was correct or not.
    • Calling it “Bear in the Cave” was the idea of a math specialist I follow and clicking on this link will take you to her site with the opportunity to get the directions and recording sheet (Math CoachsCorner:  mathcoachscorner.com Bears in the Cave freebie)
    • Be sure when students are playing that they say the number pairs outloud such as, “3 and 4 make 7.”
  4. Roll and Cover Game / Four in a Row:
    • Items needed:  A blank grid template (4×4 or larger), counters or crayons for each player (up to 3), and one of the following to create numbers needed to play (spinner, number cards, custom dice).
    • With the grid template, create the game board by randomly placing all of the numbers making up the number pairs for the focus number and fill up the grid. If working on number pairs of 6 as pictured, place these randomly:  0, 6, 5, 1, 2, 4, and 3
    • Using a spinner, custom dice, or number cards, select the first number (example 5).  Make this sentence frame:  “2 goes with ____ to make 6.”  Locate the missing number on the grid and put a counter there (or color if using a printed worksheet). How to create an easy spinner: Draw one with the number of spaces needed and duplicate for multiple students. To use, students place a pencil vertically on the center of the spinner to hold a paper clip at the center. Spin the clip.
    • The object is to try to get 4 of your counters (or colors if using a worksheet) in a row (vertically, horizontally, or diagonally).  Blocking your opponents may be necessary to keep them from getting 4 in a row.
    • A freebie attached for Number Pairs of 6 (same as picture):Capture A game of six CE
  5. Stories:  Students can create stories using pictures from clip art or other art work:

    6 children and 1 adult = 7 OR 4 girls and 3 boys = 7  Or 2 pink shirts + 5 other shirts = 7

Assessment:

  1. This page can be used to record a student’s mastery of the number pairs / bonds.  On all assessments, observe if student names hiding amount immediately (meaning fact is known) or uses fingers or other counting methods such as head-bobbing, etc. For mastery, you want the student to be able to name the missing amount quickly.Click here for free PDF copies: Number Bond Assessment by CE and Number Pairs assessment class recording sheet CE
  2. The Hiding Game above can also be used as an assessment as the teacher controls how many showing / hiding.  Ask the same questions each time:  “How many showing?”  and “How many hiding?”
  3. Folding dot cards:  Hold one flap down and open the other. Ask, “How many dots?”  Then ask, “How many hiding?”I got these free at one time from www.k-5mathteachingresources.com, but not sure they are available now. At any rate, they look easy to make.These are also good to practice with a partner.Here is a similar one I made for FREE with the PDF copy :Number Bond 3-10 assessment in part-whole format
  4. Whole-Part-Part Template:  Using a circular or square template, place a number or objects in one of the parts.  Ask student how many more are needed to create the focus number.  This can also be done with numbers only as shown in this picture.

Let us know if you have tried any of these, or if you have others that you’d like to share!  

As I’ve mentioned before, as a consultant I am available to help you as an individual, your grade level team, or your school via online PD, webinar, or just advice during a Zoom meeting.  Contact me and we can make a plan that works for you.  If you are interested in tutoring during your “spare time” check out my link for Varsity Tutors on the side bar.  Mention my name and we both get a bonus. Have a wonderful, SAFE week.  Mask up for everyone!

Number Pairs / Number Bonds Activities (PreK-2): Part 1

by C. Elkins, OK Math and Reading Lady

Learning the combinations for numbers (number pairs / numbers bonds) is critical for both operations — addition and subtraction. This is slightly different than fact families, but it’s related.  With number bonds, students learn all of the possible ways to combine 2 numbers for each sum.  Think of whole / part / part.  If five is the whole amount, how many different ways can it be split or decomposed?  For example these combinations illustrate ways to make 5:

  • 5 = 1 and 4  (also 4 and 1)
  • 5 = 2 and 3  (also 3 and 2)
  • 5 = 5 and 0  (also 0 and 5)

Knowing these combinations will aid a student’s understanding of the relationship of numbers as they also solve missing addend and subtraction problems.  For example:

  • For the problem 2 + ___ = 5.  Ask, “What goes with 2 to make 5?”
  • For the problem 5 – 4 = ____.  Ask, “What goes with 4 to make 5?”

I suggest students work on just one whole number at a time and work their way up with regard to number bond mastery (from 2 to 10). You may need to do a quick assessment to determine which number they need to start with (more of assessment both pre and post coming in Part 2). Once a student demonstrates mastery of one number, they can move on to the next. It is great when you notice them start to relate the known facts to the new ones. Here are a few activities to practice number pairs.  They are interactive and hands-on.

One more thing:  PreK and KG students could work on these strictly as an hands-on practice, naming amounts verbally.  Using the word “and” is perfectly developmentally appropriate:  “2 and 3 make 5”.  With late KG and up, they are ready to start using math symbols to illustrate the operation.

  1.  Shake and spill with 2-color counters: 

    Shake and Spill

    Use 2 color counters.  Quantity will be the number the child is working on.  Shake them in your hand or a small paper cup. Spill them out (gently please). How many are red? How many are yellow?  Record on a chart.  Gradually you want to observe the child count the red and then tell how many yellow there should be without counting them. This will also aid a student with subitizing skills (naming the quantity without physically counting the objects). To extend the activity, you can create a graph of the results, compare results with classmates, and determine which combinations were not spilled. Click on this link for the recording sheet shown:  Shake and Spill recording page

  2. Connecting cubes:  Use unifix or connecting cubes.  Quantity will be the number the child is working on. Two different colors should be available.  How many different ways can the child make a train of cubes using one or both colors?  If working with 5, they might show this:  1 green and 4 blue; 2 green and 3 blue; 4 green and 1 blue; 3 green and 2 blue; 5 green and 0 blue; or 0 green and 5 blue.  They could draw and color these on paper if you need a written response.
  3.  Ten frames: 

    Use a ten frame template and 2 different colored objects (cubes, counters, flat glass stones, candy, cereal, etc.) to show all of the cominations of the number the student is working on.  Using a virtual ten frame such as the one here Didax.com virtual ten frame or here Math Learning Center – Number Frames are also cool – especially if you are working from home or don’t want students to share manipulatives.

  4.  On and Off:  This is similar to shake and spill above.  Use any type of counters (I especially love the flat glass tones for this myself) and any picture.  For my collection, I chose some child-friendly images on clip art and enlarged each one separately  to fit on an 8.5 x 11 piece of paper (hamburger, football, flower, Spongebob, ice cream cone, unicorn, etc.).  Put the page inside a sheet protector or laminate for frequent use.  Using the number of counters the student is working with, shake them and spill above the picture.  Count how many landed on the image and how many landed off the image.  Like mentioned above, the goal is for the student to be able to count the # on and name the # off without physically counting them.  1st and above can record results on a chart or graph.  Often just changing to another picture, the student feels like it’s a brand new game!  You might also like to place the picture inside a foil tray or latch box to contain the objects that are dropped.  The latch box is a great place to store the pictures and counters of math center items.
  5.  Graphic organizers:  The ten frame is a great organizer as mentioned earlier, but there are two whole/part/part graphic organizers which are especially helpful with number pairs – see below.  Students can physically move objects around to see the different ways to decompose their number.

Check out Jack Hartman’s youtube series on number pairs from 1 to 10. Here’s one on number pairs of 5:   “I Can Say My Number Pairs: 5″ He uses two models (ten frames and hand signs) and repetition along with his usual catchy tunes.

Also, please check out the side bar (or bottom if using a cell phone) for links to Varsity Tutors in case you are interested in doing some online tutoring on the side or know students who would benefit from one-on-one help. Please use my name as your reference — Cindy Elkins.  Want some PD for yourself?  Contact me and I’ll work out a good plan to fit your needs!

Next post:  More activities for learning number bonds and assessment resources (both pre- and post-).  Take care!!

 

Number Talks – Online

by C. Elkins, OK Math and Reading Lady

You know I am a huge advocate of doing daily number talks. I have written several posts about this which I will link below.  But how can you conduct a number talk via Zoom or whatever platform you are using?  Here are some suggestions.

  1. Post a problem on your screen. Write it horizontally (so as not to immediately suggest it should be solved via the standard algorithm).
  2. Ask students to show a way they might solve the problem.  Using a marker (so the end product will show up when displayed), students work on their whiteboards or notebook paper tablet.
  3. Give a reasonable amount of time (depending on the grade level and the problem given).  Teacher can even play some soft background music to signal time to start working.
  4. Students signal with a thumbs up when they are done (on their screen or in the chat box).
  5. The teacher can interject he/ she would love for some of the students to share their thinking, so when they are done and waiting for the others, think mentally on how they might explain it.
  6. With a signal to end working time, students then hold up their whiteboards.
  7. The teacher can select some to share (or students can volunteer) showing the different strategies used.  The teacher can model the strategy on his/her screen as the student verbally describes it.
  8. Different strategies can be recorded on an anchor chart for future reference.

    Here are some links from my Number Talks posts.

Professional Development Opportunity

As you know, I have been working as an educational consultant the past five years — job-embedded professional development with elementary teachers regarding math and reading instructional strategies. With the COVID-19 nightmare, schools are closed in most locations. School administrators are hesitant to commit to job-embedded consultants right now because there are so many uncertainties.  However, if you as a teacher or parent are interested in private one-on-one online consultation visits with me, I am available to help you reach your instructional goals.  We will work out a plan that is easy on your budget and schedule. Contact me via the comment box with a brief request and I will email you privately.

What can we work on?

  • Reading strategies (phonemic awareness, phonics, cueing and prompts, comprehension, text structures, fluency . . .)
  • Math strategies (subitizing, number sense, addition, subtraction, multiplication, division, place value, rounding, fractions, geometry, . . .)
  • Interpreting data
  • Writing and spelling
  • Other topics you don’t see here?  Just ask.

Tutoring Opportunities

If you know students who are in need of online tutoring (anywhere in the US at any grade level PreK-College), you are invited to refer them to Varsity Tutors using my name (Cindy Elkins).  It is a very reputable company that matches tutors with students in any subject or grade level. https://www.varsitytutors.com?r=2Asn3c

If you are interested in becoming an online or in-person tutor yourself, you are also invited to contact Varsity Tutors. You would be an independent contractor who can set your own hours and accept only the students you feel comfortable working with. Payments are direct deposited twice a week. Give them my name please. Use this link: https://www.varsitytutors.com/tutoring-jobs?r=2Asn3c

Click on the badge icon with my photo on the right sidebar to check them out. Or the links above. On your phone app, the badge will be at the bottom.

**I do receive a bonus when my name is used as a referral. Thank you for your trust in me!

Stay safe everyone!  

 

24 Summer Time Math Activities which can be done at home!

by C. Elkins, OK Math and Reading Lady

I realize many of you  (teachers and parents) may be searching for ways to link every day activities to math so that children can learn in a practical way while at home during this surrealistic period.  Happy Fourth of July and . . . .Here’s a list of things you might like to try:

 

 

 

 

Outdoors

  1. While bouncing a ball, skip count by any number. See how high you get before missing the ball. Good to keep your multiplication facts current.
  2. How high can you bounce a ball? Tape a yardstick or tape measure to a vertical surface (tree, side of house, basketball goal). While one person bounces, one or two others try to gauge the height. Try with different balls.  Figure an average of heights after 3-4 bounces.
  3. Play basketball, but instead of 2 points per basket, assign certain shots specific points and keep a mental tally.
  4. Get out the old Hot Wheels. Measure the distance after pushing them.  Determine ways to increase or decrease the distance. Compete with a sibling or friend to see who has the highest total after 3-4 pushes.  Depending on the age of your child, you may want to measure to the nearest foot, inch, half-inch or cm.
  5. Measure the stopping distance of your bicycle.
  6. Practice broad jumps in the lawn. Measure the distance you can jump. Older students can compute an average of their best 3-4 jumps. Make it a competition with siblings or friends.
  7. Some good uses for a water squirt gun:
    • Aim at a target with points for how close you come. The closer to the center is more points.
    • Measure the distance of your squirts. What is your average distance?
    • How many squirts needed to fill up a bucket?  This would be a good competition.
  8. Competitive sponge race (like at school game days): Place a bucket of water at the starting line. Each player dips their sponge in and runs to the opposite side of the yard and squeezes their sponge into their own cup or bowl. Keep going back and forth. The winner is the one who fills up their container first. Find out the volume of the cup and the volume of water a sponge can hold.
  9. Build a fort with scrap pieces of wood. Make a drawing to plan it. Measure the pieces to see what fits. Use glue or nails (with adult supervision).
  10. Take walks around the neighborhood. Estimate the perimeter distance of the walk.
  11. In the pool:
    • Utilize a pool-safe squirt gun (as in #6 above).
    • Estimate the height of splashes after jumping in.
    • Measure the volume of the pool (l x w x h).  The result will be in cubic feet.  Convert using several online conversion calculators such as this one: https://www.metric-conversions.org/volume/
    • Measure the perimeter of the pool.  If it is rectangular, does your child realize the opposite sides are equal.  This is a very important concept for students regarding geometry (opposite sides of rectangles are equal).
    • What if you want to cover the pool? What would the area of the cover be?
    • Measure how far you can swim.  Time the laps.  What is the average time?
  12. Watch the shadows during the day. Notice the direction and the length.  Many kids don’t realize the connection between clocks and the sun. Make your own sun dial. Here are a few different resources for getting that done, some easier than others:

 

Indoors

  1. Keep track of time needed (or allowed) for indoor activities:  30 minutes ipad, 1 hour tv, 30 minutes fixing lunch, 30 minutes for chores, etc.  This helps children get a good concept of time passage. Even many 4th and 5th graders have difficulty realizing how long a minute is.  This is also helpful as a practical application of determining elapsed time. Examples:
    • It’s 11:30 now.  I’ll fix lunch in 45 minutes. What time will that be?
    • I need you to be cleaned up and ready for bed at 8:30.  It’s 6:30 now.  How much time do you have?
    • You can use your ipad for games for 1 hour and 20 minutes.  It is 2:30 now. What time will you need to stop?
  2. Explore various recipes and practice using measuring tools.  What if the recipe calls for 3/4 cup flour and you want to double it?
  3. In the bathtub, use plastic measuring cups to notice how many 1/4 cups equal a whole cup. How many 1/3 cups in a cup? How many cups in a gallon (using a gallon bucket or clean, empty milk carton)?
  4. While reading, do some text analysis regarding frequency of letter usage.
    • Select a passage (short paragraph).  Count the number of letters.
    • Keep track of how often each letter appears in that passage. Are there letters of the alphabet never used?
    • Compare with other similar length passages.
    • After analyzing a few, can you make predictions about the frequency of letters in any given passage?
    • How does this relate to letters requested on shows such as “Wheel of Fortune” or letters used in Scrabble?
  5. Fluency in reading is a measure of several different aspects:  speed, accuracy, expression, phrasing, intonation.
    • To work on the speed aspect, have your child read a selected passage (this can vary depending on the age of the child). Keep track of the time down to number of seconds. This is a baseline.
    • Have the child repeat the passage to see if the time is less.  Don’t really focus on total speed because that it not helpful for a child to think good reading is super fast reading. Focus more on smoothness, accuracy and phrasing.
    • Another way is to have a child read a passage and stop at 1 minute. How many words per minute were read?  Can the child increase the # of words per minute (but still keep accuracy, smoothness, and expression at a normal pace)?
  6. Play Yahtzee!  Great for addition and multiplication.  Lots of other board games help with number concepts (Monopoly, etc.)
  7. Lots of card games using a standard deck of cards have math links. See my last post for ideas.
  8. Measure the temperature of the water in the bathtub (pool thermometers which float would be great for that). How fast does the temperature decrease. Maybe make a line graph to show the decline over time.
  9. Gather up all of the coins around the house.  Read or listen to “Pigs Will be Pigs” for motivation. Keep track of how much money the pigs find around the house. Count up what was found. Use the menu in the back of the book (or use another favorite menu) to plan a meal. Be sure to check out Amy Axelrod’s other Pig books which have a math theme Amy Axelrod Pig Stories – Amazon  Here is a link to “Pigs Will be Pigs”: Pigs Will Be Pigs – Youtube version
  10. Help kids plan a take-out meal that fits within the family’s budget.  Pull up Door Dash for a variety of menus or get them online from your favorite eateries. This gives great practical experience in use of the dollar to budget.
  11. Look at the local weekly newspaper food advertisements.  Given a certain amount of $, can your child pick items to help with your shopping list?  If they accompany you to the store, make use of the weighing stations in the produce section to check out the weights and cost per pound.
  12. Visit your favorite online educational programs for math games or creative activities.  See a previous post regarding “Math Learning Centers.” The pattern blocks and Geoboard apps allow for a lot of creativity while experiencing the concept of “trial and error” and perseverance. These can be viewed at the website or as an app.  Here’s a link to it to save you time. Virtual math tools (cindyelkins.edublogs.org)

Please share other activities you recommend!!  Just click on the speech bubble at the top of this post or complete the comments section below.  I miss you all!

Helpful reading and math aids for parents

by C. Elkins, OK Math and Reading Lady

With so many parents taking on the role as teacher, I thought I would provide some resources you can pass along to parents.  In this post you will find some reading strategy help via one-page parent friendly guides (for primary and intermediate). I also included resources for math to provide some practical activities at home as well as some fun card and dice games that emphasize math skills.  Feel free to pass them along. Enjoy!

Reading

Math

On another note:

I am in the process of moving from OK to Arizona!! We have lived in our home for the past 35 years . . . but we have this opportunity to be closer to our family (two sons, a daughter-in-law, our only grandson, and my sister).  I am taking all of my teaching materials with me and still plan to continue my blog, develop more instructional resources, and provide PD via online platforms. I hope you all will stay tuned!!  Stay safe everyone!!

 

Telling Time Part 4: Elapsed time (continued)

by C. Elkins, OK Math and Reading Lady

In my last post, I shared my favorite model for elapsed time (Mountains, Hills, and Rocks) using an open number line. In this post I will share another version of the # line some of you might like — I’ll list the pros and cons of it as well as show the std. algorithm / convert version.

I hope all of you are doing well. I realize many of you are involved in distance learning with your students – and this may be in addition to taking care of your own children’s needs at home. So I understand my blog might not be on your top list of priorities, but I do hope you will bookmark it and keep it for future reference.  But again, if you are home with kid, then dealing with elapsed time is a perfect real-life math situation they can apply on a daily basis.

The Z Model:

The Z model is a straight number line “bent” into 3 parts of the Z:

  • 1st “leg”:  From start time to next full hour – determine how many minutes
  • 2nd “leg”: From hour to hour – determine how many whole hours
  • 3rd “leg”: From last hour to end time – determine how many minutes

Here is an example to see the steps involved:

Here’s another in one single view to determine elapsed time between 7:50 and 1:10:

Pros:

  1. It helps break time down into smaller chunks.
  2. It’s a visual model which can help a child mentally process the elapsed time in these chunks.

Cons:

  1. Students would more likely have to know automatically how much time has elapsed on the first “leg” of the Z. In other words, can they mentally figure that the time from 8:25 to 9:00 (the nearest hour) is 35 minutes?  Or the elapsed time from 3:47 to 4:00 is 13 minutes?
  2. In my opinion, this model is mostly just helpful when start and end times are given and the task is to compute the total elapsed time. It would not be very helpful if the task was to determine the start or end time.
  3. If a child can figure the minutes of elapsed time of the first “leg” of the Z, they might not need the visual model to solve.

The Std. Algorithm / Converting Model

This model resembles a std. algorithm problem because time is aligned vertically and added or subtracted.

  • When adding, any minutes which total 60 or over would be converted to hours.
  • When subtracting, exchange 1 hour for 60 minutes.

Here is an example to see the progression from start to finish when start time and elapsed time are the known parts:

Here’s another example in one view:

Contextual scenario: At 2:45 I went to the zoo. We stayed there 3 hours and 25 minutes. What time did I leave the zoo?

Here’s an example that involves a known end time and elapsed time. The problem is to determine the start time which involves subtracting time:

 

And another problem in one view:

Before I got ready for bed at 9:20 p.m., I spent 2 hours and 35 minutes doing homework. What time did I start my homework?

Pros:

  1. Students who are ready for more abstract strategies might enjoy this model.
  2. This model is more useful when solving problems in which the task is to find the end time or start time.
  3. This can be utilized with hours, minutes, and seconds problems.

Cons:

  1. Having strong knowledge of number combinations that equal 60 is needed.
  2. There may be two or three steps involved to arrive the final answer.
  3. The regrouping in the subtraction version may involve two types which could be confusing: minutes vs. base 10 (as shown in picture directly above)
  4. Understanding what converting time means and why we subtract and add within the same problem (subtract 60 minutes, but add 1 hour).

I miss seeing my friends in person!  Let me know how you are coping during these crazy times!

 

Telling Time Part 3: Elapsed Time – Start and end time known

by C. Elkins, OK Math and Reading Lady

Wow! What a difference a couple of weeks makes.  My last post (Time Part 2) was 2 weeks ago, and life was pretty normal here then. Maybe you are using your extended time off to just try to calm down, maybe you are catching up on home chores or your favorite Netflix series, or maybe you are digging out some favorite recipes. Just in case you are using this time to help your own children with learning objectives or catching up on some PD for yourself, I am here to help any way I can. Remember in the black bar above you can access my learning aids without reading all of the articles to find them. Or type what you are looking for in the search box. Or look at the categories list to pull up by topic.

Today’s post will focus on concepts related to elapsed time.

As I mentioned in Telling Time Parts 1 and 2, it is important for students to have a concept of time. How long is a second, a minute, an hour? What tasks can be accomplished in those amounts of time. These are foundational concepts students need to better understand elapsed time. Are you making notes of time during the school day (or at home now) to make it relevant?  Questions or statements such as these are helpful:

  • “We have 10 minutes to finish attendance and lunch count. Look at the clock so you can keep track.”
  • “Lunch will be ready at 12:00 noon.  Look at the clock. It’s 11:30 now, so lunch will be ready in 30 minutes.”
  • “It’s time to get ready to go home. Look at the clock. What time is it?  You should be ready in 5 minutes. What time will be it then? What will the clock look like?”
  • To help speed up time for transitions and work on a class management goal at the same time, try this for a procedure such as lining up: “Boys and girls, it’s time to line up to go to PE.” As students line up, you as the teacher will silently keep track of how much time it takes students to get ready. When they are ready, say someting like:  “It took you 3 minutes 20 seconds to get ready. We miss learning time when it takes this long.  Let’s see if we can beat that next time.”   Most kids respond well to this mini challenge.  If it’s a real contentious issue in your class, this can be followed with an easy reward such as: “It took 3 min. 20 seconds to line up and get ready. That is too long. Next time we line up if you can get ready in less than that time, I will keep track by building the word G-A-M-E.  You earn a letter each time you beat the previous time to line up.  The time starts when I say line up and the time stops when everyone is facing the front, quiet, and hands to themselves. You must walk to do this.”  Building a short word helps students earn a reward in a short amount of time so they are more likely to strive to meet the goal. It is easy to implement and can easily be incorporated into a reading or math game.  The word to build could also be F-U-N.  Then it’s wide open to what that could be:  A video, talk time, drawing time, a few minutes extra recess.  Yes, this takes time also – but it helps students work together toward a common goal, and may save your sanity.  This “time” technique can also be applied to other procedures such as getting out materials, staying quiet, etc.  One hint:  Don’t do a countdown or let students know how much time they are taking as you are keeping track.  If you announce, “We are at 2 minutes . . . you might make it.” this gives students knowledge they have time to waste.” We are trying to build an awareness of time along with a sense of urgency and teamwork. So wait until they are all ready to announce the time it took.

Okay, a little off topic – but showing how there are many ways to help students become more aware of time in their daily lives.

As in most story problems related to time, there are 3 components.  The story gives 2 of them, and the problem is to find the missing one:

  1. Start time
  2. Elapsed time – the time it takes for something to be finished
  3. End time

There are several common strategies, some which are more pictorial and some which are more abstract.  Of course, I am in favor of those which provide some visual representation at first such as an open number line or a Z-chart. I will feature the number line model today.  More abstract models are the T-chart and lining up times vertically like you would doing a standard algorithm and adding / converting times. I’ll focus on those in future posts.

Number line:  There are a few versions of time number lines out there which help students move from start time to end time. Some already have time increments noted on the line, some use jumps that all look the same.  I happen to love the “Mountains, Hills, and Rocks” look because it helps immediately to differentiate between the hours and minutes and doesn’t require any advance preparation as with pre-marked number lines. The mountains represent hours, the hills increments of minutes, and rocks are individual minutes.  I will share 3 types of elapsed time problems, but just elapsed time unknown in this post:

  • elapsed time unknown
  • end time unknown, and
  • start time unknown

Elapsed time unknown: This features stories in which the start and end time are given.  So students must find the elapsed time. Bobbi went to the movie theater at 7:15 p.m.  It ended at 9:45 p.m.  How long did the movie last?

    • Put the known parts on the number line and label  (start at 7:00 / end at 9:45).
    • Underline the hour part of the number.  Can we add an hour to the 7? Yes.  What time would it be then? 8:15.  Now here is how we show an hour (with a mountain). Can we add another hour? Yes. What time would it be then? 9:15. Add another mountain and keep track of the time under the line.  Can we add another hour? No. Why not? It would be 10:15 which is past the end time.
    • Now we will switch to minutes (called the hills).  The hills are used to show increments of 5, 10, 15, 20, 30, etc. Since we all write different sizes, etc., I continuously tell students this:  “It’s the number we write inside of the hill that matters more than the size or length of the hill.”  This is because sometimes due to space limitations, my 5 min. hill looks the same length as my 10 minute hill.
    • Underline the minutes part of the number 9:15.  Now let’s add minutes until we get to 9:45.  This can be done several ways depending on students’ understanding.  I might make hills of 5 minutes each.  In this problem, I might make hills of 15 minutes each.  I might want to add 5 minutes in one jump to get my minutes to a number ending in 0. Some students would realize that 30 minutes would connect us from 9:15 to 9:45.  When teaching and modeling, we all do the same way. Then when they seem comfortable, we look at different ways to show the same problem.  This provides a safety net for some, while a challenge for those who enjoy it. For this example, I would say: “Let’s get our 9:15 to an easier time to work with . I’m going to just add 5 minutes. Looking just at the minutes part of the number, what is 15 + 5??” Yes, 20. So what time would it be now? Yes, 9:20. Our number now ends in a zero, which we can add to mentally. Let’s add 10 min. to that. What is 20 + 10? Yes, 30. So what time is it now? Yes, 9:30. Let’s add another 10 minutes. Can we do that? Yes, because 30 + 10 is 40 and 9:40 is before 9:45. Now how much time is there between 9:40 and our end time of 9:45? Yes, just 5 minutes. So that will connect us to the end time of the movie at 9:45, and we are almost done!
    • The last step is to look at the numbers we wrote inside our mountains and hills and combine them. You will see Bobbi was at the movie theater for 2 hours (2 mountains) and 30 minutes (5 + 10 + 10 + 5) = 2 hours, 30 minutes.

Stay tuned for more examples of elapsed time problems through the next few posts. Future posts will provide some freebie story problem practice and good resources you might like.              And stay safe and well!!!

Telling Time Part 2: Reading a clock

by C. Elkins, OK Math and Reading Lady

In this post, I will present some ideas for reading and drawing clock times (especially the analog):  to the hour, half hour, quarter hour, and 5 minute increments. Along with practice telling time should be opportunities to put it in context.  For example, While setting the clock for 8:00,  I would mention that at 8:00 in the morning I might be getting ready for school or eating breakfast, while at 8:00 at night I might be doing schoolwork, watching tv, or getting ready for bed.  Look for some freebies throughout this post!

Time to the hour:

  1. Short hand / short word = hour
  2. Long hand / longer word = minutes
  3. Use an anchor chart to show a large clock and label the hands.
  4. Always look for the short hand first when naming time to the hour.
  5. Show with a Judy clock or the clock on https://www.mathlearningcenter.org/resources/apps/math-clock. Observe what happens to the hour hand when the minute hand moves all the way around the clock one time.  Admittedly, this is a hard concept for kids because we are imitating an hour in time in only a few seconds. And no one has time to watch the clock for an hour!!
  6. Discuss what events take about an hour to accomplish (see Telling Time Part 1 for more info).
  7. Draw pictures to show 2 different times of day with the same time (8:00 in the morning, 8:00 in the evening)
  8. If you want students to practice drawing a clock correctly with the hour notations, try it in the steps shown above.

Time to the half-hour:

  1. Shade half of the clock
  2. Show the position of the hour hand when it is half-past the hour.  It should be positioned half way between the 2 hour numbers.  I usually show students you should be able to tell the approximate time even if the minute hand was missing based on where the hour hand was located between two numbers.  So when students are drawing hands to show 7:30, help them see the hour hand will be half-way between the 7 and the 8.
  3. Brainstorm events which take about 30 minutes to accomplish.
  4. During the morning or afternoon, announce each time 30 minutes has passed.

**For 3rd and up, start looking at what 30 minutes of elapsed time looks like on a clock.  The minute hand will be directly opposite where it started out.  For example:  3:40 + 30 minutes = 4:10.  The minute hand would change from the 8 to 2 which cuts the clock in half.

Time to the quarter hour:

  1. Practice dividing a circle into fourths (vertical and horizontal lines) and label with 12, 3, 6, and 9.
  2. Label your classroom clock with 15, 30, and 45 next to the 3, 6, and 9. Here’s a freebie from “Dr. H’s Classroom” on TPT: Clock labels – FREE
  3. 15 minutes is a fourth of 60.  Or 15 + 15 + 15 + 15 = 60.  Check that students aren’t confusing it with 25 minutes (since 25 cents is a quarter dollar).  Remind students that “quart” is common in many terms:  quarter (4 in a dollar); quartet (4 singers); quart (4 of them in a gallon); quarter in sports (one fourth of the game).
  4. Brainstorm events that might take about 15 minutes to accomplish.
  5. Check out my Time to the Quarter Hour lesson practice and game below.
    • 8:15 — eight fifteen, quarter past 8, 15 past 8
    • 8:30 — eight thirty, half past 8, 30 minutes past 8, 30 minutes until 9:00
    • 8:45 — eight forty-five, 45 minutes after 8, 15 minutes til 9:00

Here are two FREE activities to practice time to the half hour and quarter hour.

  • The first one is a guided practice to help students learn different ways to write the same time. I usually have them select 2 ways from the options at the top (or bottom). The packet includes time to the half hour, quarter after and quarter til, sample answer responses, and a blank page to create your own. Click HERE
  • One is a game I named “Tic-Tac-Time.”  Students play with a partner on a clock tic-tac-toe board.  I provided a black print version and a color version. For the spinners page, students will need a paper clip or if you have clear spinners to place over top, that is great! Students spin the time using both spinners, then pick a spot on the tic-tac-toe grid to help them potentially get 3 in a row. They draw in the hands and write the time. Click HERE for that game.

Time to five minutes:

  1. The key, of course, is counting by 5s as you go around the clock.  But do students always start at 12 and count all the way around no matter where the minute hand is positioned?  Perhaps if the minute hand is at the 8, they can start with 30 (at the 6) and count 35, 40 to the 8.
  2. Brainstorm events that might take about 5 minutes to accomplish.
  3. Again, make sure students look at the hour hand first, then the minute hand.

My pet peeve about drawing clock hands: I usually insist students just draw straight lines or use very small arrows on the clock hands because they often put huge arrows at the end that are distracting (and time consuming).  We also practice the length of each hand such as this:

  • Minute hand extends from the center to the edge of the clock
  • Hour hand extends from the center to just touch the number

What are your favorites for helping kids tell time correctly? Please share!

 

Telling Time Part 1: Basic concepts

by C. Elkins, OK Math and Reading Lady

Concepts of time are one of the subjects we teach at school, but often has more application at home:  Time for bed, time to eat, time to clean up your room, time to play, and so on. I have found when working with students in 3rd and 4th grades about elapsed time, that they often don’t have a very good concept of time. It’s no wonder. We (as teachers or parents) say, “You have 1 minute to . . .” or “I’ll be there in a minute!”  But in reality that minute has stretched to much more like 5 or 10.

So what can we do to help with concepts of time at school (or home)?

  • Post a copy of the daily schedule. Refer to it often.
  • Use a timer for certain tasks.
  • If you announce a time, stick to it.

Try these activities with students. The ones you use will depend on the grade level. Click here for a FREE copy of the brainstorming recording sheets (pictured below): What can you do in 1 sec., min., hour

  1. “Tick-tock” — It takes about 1 second to say this word.  Brainstorm what things can be done in this amount of time. Try some of them out (clap, blink, snap, swallow, etc.). It’s effective and engaging to have students brainstorm first with a partner before sharing with the whole class.
  2. Watch an analog clock for 1 minute:  Observe the second hand going around 1 complete time. It feels like a long time has passed when actually watching it. Brainstorm things that can typically be completed in one minute (brush teeth, put on socks and shoes, drink some water, etc.)
  3. You may want to discuss other chunks of time (especially 5 minutes or 15 minutes since we eventually want students to be able to read a clock in these increments). 5 minutes — eat a snack, get dressed, walk across the school.  15 minutes — walk to school, finish a worksheet, eat a sandwich.
  4. Brainstorm events that take about 30 minutes (eat lunch, watch a sit-com, take a bath) and an hour (basketball practice, chores, shopping, math period).
  5. Incorporate writing and drawing to name a start time and an end time with a label or a couple of sentences about the activity (see attached). Even 1st and 2nd graders can begin to think about this amount of elapsed time.

Once students have a better understanding of how long something takes to finish, then students will have a better grasp of telling time and determining reasonableness of elapsed time problems. Plus it may enable them to become better judges of their own time with regards to home chores and school assignments and events.

Enjoy your week! Time Part 2 coming next.

 

 

 

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!