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!

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.