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?:
Students gain the ability to subitize (tell a quantity without physically counting).
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.
Students gain the ability to see many different ways a number can be composed or decomposed which assists with addition and subtraction facts.
Students gain practice with strategies such as counting on, add/subtract 1, doubles, near doubles, adding 9, adding 10, missing addends, and equal groups.
Teachers are able to observe students’ processing skills in an informal math setting.
You can create your own cards with dot stickers (or bingo stamps) on large index cards or paper plates
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?
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.
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.
The teacher observes to see who is counting, who is participating, who uses fingers, who is quick /slow, etc.
Teacher asks random students, “How many dots?”
Teacher asks random students, “How did you see them?”
Results can be stated verbally or written down by the teacher.
Here are some examples with sums less than 10:
How do you see 9?
How do you see 8?
I can imagine 1 counter moving to help me think of 4 + 4 (bottom image).
Here are examples using 2 ten frames to illustrate quantities greater than 10:
I try to look for doubles (6 + 6).
I see another use of doubles (5 + 5)
I use the near doubles strategy to think 6 + 6 and add one more since 7 is one more than 6. I could also try 7 + 7 and then take one off.
Next post: I will feature ways to use the random dot cards for your Number Talk sessions.
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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.
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!
6 x 3 (six groups of 3)
3 x 6 (three groups of 6)
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.
12 x 4 (12 groups of 4)
4 x 12 (4 groups of 12): or 4 x 10 plus 4 x 2
Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.
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?
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
Start with 14 (red)
Turn over 9 (yellow)
Move the red together
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.
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 →
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: