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

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.

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:

• If you find you don’t have enough base ten blocks to go around, then the mini ten frames might be a good substitute for students to show their understanding of place value regarding tens and ones.
• Provide individual students (or pairs of students) a baggie of prepared tens frames. You would need 10 complete 10 frames and then 1 of each of the others (0-9).
• Using a blank tens/ones mat, state a 2 digit number such as 35.  Show students the 3 represents how many 10’s there are (3) and the 5 represents how many 1’s (5). Place ten frames on the mat to illustrate.

35 — With mini ten frames

• Vary how you ask students to show amounts:  Sometimes say, “Show me 35.” vs. “Show me 3 tens and 5 ones.” vs. “Show me 30 and 5.”

Variation of using base ten blocks with this place value mat:

• This place value mat (link above in opening paragraph) allows you to use your base ten blocks on the ten’s side and the units cubes on the one’s side, with one helpful difference.
• The ten frame template allows students to organize their ones as opposed to random placement when no ten frame is present. This helps students with number bonds and it really helps you, as the teacher who is observing, to determine immediately if the student placed the correct # of units.*
• With the examples shown (47), students can show the ones as 5 + 2 or 4 + 3.

*Even without the use of this mat with printed ten-frame, I insist students show some type of organized placement of units cubes any time they are being used for some type of counting. Students can be creative with patterns that resemble domino or dice dots, ten frame configurations, equal rows, etc. Try it!!!

The ones cubes are organized!

Adding or Subtracting 2 digit numbers:

• Use a generic tens/ones mat and the mini ten frames so students can model problems such as these:
• Use the tens/ones mat (with ten frames). Utilize both ten frames in the one’s place for adding two 2-digit numbers.
• Students can use units cubes or counters for the one’s place for concrete experiences.

This shows 64 + 19

• Utilize some of the previously mentioned strategies for working with doubles, near doubles, 9, etc.
• Show regrouping: Example 82 – 7.  Start with 82 (with all purple tens on the ten’s side). Since there aren’t enough ones to subtract 7, regroup by moving ten to the one’s place (shown in picture below).  Critical step:  Be sure to have students see that there are still 82 dots on the board (70 + 12). Now 7 can be removed (2 from the orange card and 5 from the purple card, which leaves 5). The answer would be 75.
• For pictorial practice, laminate the mats and students can draw in the pieces with dry-erase markers.

Counting coins:

Check out this free resource from one of my favorite math specialists (Math Coachs Corner):

Coin identification and value activity with ten frames

Have a great week! Let us know how using ten frames has helped your students!

# Ten Frames Part 2: Addition and subtraction

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. 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.

• Place counters on the ten frame. Determine how many more are needed to fill in the ten frame. This also helps with missing addends.  Example:  3 + ___ = 10.  Ask, “What goes with 3 to make 10?”
• Using 2-color counters, fill the 10 frame with 1 color. Then turn over some to reveal a number bond of 10 (such as 4 and 6).

Shake and Spill

• Play “Shake and Spill” with 10 two-color counters.  Click on these links for Shake and Spill Directions and a Shake and Spill recording page. Basically, the student puts 10 of these counters in a cup, shakes it, and spills it out (gently). Count how many red and how many yellow. Repeat 10 or more times. Keep track of the spills on a recording sheet. Which combination came up most often? Which combination never came up? What is really nice to observe is if a student spills counters and sees  6 are red, do they know automatically there are 4 yellow, or do they still have to count them?
• Since number bonds enable a student to see addition and subtraction problems, the second bullet above will serve subtraction problems very well.  Start with 10, turn over 7 to the yellow side. How many counters are red?

Make a Ten: This strategy builds on the above (facts of 10) to help with problems with sums between 10 and 20. Students should readily be able to solve a problem such as 10 + 4 mentally first.

• Use 2 ten frames (see Ten Frames part 1 for a link for templates)
• Let’s say the problem was 8 + 5.  Place 8 counters on one ten frame, place 5 on the other.
• Move counters from one ten frame to fill up the other.  8 + 5 is the same as 10 + 3. The problem 10 + 3 should be a mental math problem. Students will need to see that counters were not added, but shifted from one ten frame to the other.
• Repeated practice with this concrete activity helps children think more deeply about the relationship of numbers.

Continued practice with these strategies:

1. During your daily math meeting, flash ten frame dot cards to students in which they must use the above strategies. Use it as a # Talk sessions so students can verbally explain how they solved it.
2. Try this from NMCT Illuminations sight (National Council for Teachers of Mathematics): Interactive ten frame
3. Watch this video of a teacher modeling the Make-a-Ten strategy: Make a ten video using ten frames
4. Learning stations:
• With a blank ten frame, create doubles and near doubles problems. Or look at flash cards and make those problems.
• Show a partially filled in ten frame. Student must tell their partner how many more are needed to fill it in.
• Give students flash cards for problems with answers between 10 and 20. Show each addend on a ten frame and use the make-a-ten strategy.

Share your experiences with ten frames!

# Ten Frames Part 1: Number Sense

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:

• Provide a blank ten frame to students along with some counters (beans, cubes, bears, cheerios, two sided counters, etc.).  State a number to count and place on their ten frame. This is a much better approach in my opinion than asking studens to randomly place counters on a blank mat (which is what Saxon advises in their KG counting lessons). Random placement means the student might easily miscount and the observing teacher cannot often tell at a glance if the student has the correct amount. OK – that’s my soapbox.
• This method allows the teacher to readily determine if the student counted correctly. It also leads to helping students see there are different ways to represent this amount (number bonds).
• The teacher can now ask students to show (and/or tell) their result. This is what the process standard of communication is all about! If most students show only 1 way, the teacher asks, “Now, can you show 6 in a different way?”
• Use a blank ten frame as part of your daily math meeting time. Select a number of the day or number of the week. Show a way to make that amount. Connect with numbers such as:  2 and 2 is 4 (PreK or KG) or 2 + 2 = 4 (late KG, 1st and up)

Learning station ideas for subitizing and number bonds with ten frames:

1. Match # cards to ten frames (use mini ten frames in resources above).
2. Provide # cards, 3-4 blank ten frames, and counters. Student turns over a # card and uses counters to show different ways to make the same amount. This physical concrete method is recommended for preK and KG. As an extension for first or second graders, they can start with the concrete and then record their responses (pictorial method) on blank mini ten frame templates.
3. Play with a partner:  Materials — one large blank ten frame per student, counters, set of # cards, and a screen between the 2 players. Turn over 1 number card that both can see. Each student makes that amount on their own ten frame (hidden from view from their partner by the screen). Then remove the screen and compare results.
4. Put ten frame dot cards in order (least to greatest, or greatest to least).
5. Play “war” with ready-made ten frame dot cards. Students start out with an equal stack of cards. Each student turns over 1, tells how many and determines who has more (or less).
6. Play “Go Fish” with mini ten frame cards. This means you will need some cards that have different ways to show the same amount. I will be on the lookout for some!! If you know of some, please share your link.

Tell us how you use ten frames to build number sense!! Or if you try doing any of the above, what were the results?

Have a great week!

# Multiplication Concepts Part 4: Skip Counting

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

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

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

Ten teaching strategies for skip counting:

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

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

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

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

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

For my Lawton, OK friends who are implementing Saxon Math but don’t have wall space for all of the math meeting components, here are a few links from TPT for grades KG-3rd for ppt / SMARTBOARD versions.  The important aspect of the morning meeting is the interactive part where individual students have a role in placing the numbers, days, coins, patterns, clock hands, graph piece, etc. on the board each day. With that said, there is also value in having some of the math meeting components visible throughout the day such as the calendar, 100 chart, and number of the day. Think of the components you or your students would most likely refer to outside of the math meeting time to keep a permanent physical display.

Here are the links. Read the other purchasers’ comments and look at the previews to get more info.  I have not purchased any of these so I can not vouch for the quality or usefulness. For those of you who already have a math meeting ppt that you recommend, please let us know!! Thanks!

KG-2nd:

# Addition and Subtraction Part 2: Part-Part-Whole Models KG-2nd

In Part 1 I focused on a numerical fluency continuum, which defines the stages a child goes through to achieve number sense. After a child has a firm grasp of one-to-one correspondence, can count on, and understands concepts of more and less, he/she is ready to explore part-part-whole relationships which lead to the operations of addition and subtraction. That will be the focus of this post. Read on for free number bond activities and a free number bond assessment!

One way to explore part-part-whole relationships is through various number bonds experiences.  Number Bonds are pairs of numbers that combine to total the target or focus number. When students learn number bonds they are applying the commutative, identity, and zero properties. Do you notice from the chart below that there are 4 number bonds for the number 3; 5 number bonds for the number 4; 6 number bonds for the number 5, etc? And . . . half of the number bonds are actually just the commutative property in action, so there really aren’t as many combinations for each number to learn after all.

• KG students should master number bonds to 5.
• First graders should master number bonds to 10.
• Second graders should master number bonds to 20.Teaching Methods for Number Bonds
• Ideally, students should focus on the bonds for one number at a time, until mastery is achieved. In other words, if working on the number bond of 3, they would learn 0 and 3, 3 and 0, 1 and 2, 2 and 1 before trying to learn number bonds of 4. See the end of this post for assessment ideas.

• Ten Frame cards: Use counters to show different ways to make the focus number. (See above example of 2 ways to show 6.) Shake and Spill games are also great for this:  Using 2-color counters, shake and spill the number of counters matching your focus number.  See how many spilled out red and how many spilled out yellow.  Record results on a blank ten-frame template. Repeat 10 times.
• Number Bond Bracelets: Use beads and chenille stems to form bracelets for each number 2-10.  Slide beads apart to see different ways to make the focus number.
• Reckenreck: Slide beads on the frame to show different combinations.
• Part-Part-Whole Graphic Organizers:  Here are two templates I like. Start with objects matching the focus number in the “whole” section. Then move “part” of them to one section and the rest to the other section. Rearrange to find different bonds for the same focus number. Start students with manipulalatives before moving to numbers. Or use numbers as a way for students to record their findings.

Once students have a good concept of number bonds, these part-part-whole organizers are very helpful when doing addition and subtraction problems (including story problems) using these structures: Result Unknown, Change Unknown, and Start Unknown.  Children should use manipulatives at first to “figure out” the story.

• Here is an example of a change unknown story:  “I have 5 pennies in one pocket and some more in my other pocket. I have 7 pennies all together. How many pennies in my other pocket?” To do this, put 5 counters in one “part” section. Count on from 5 to 7 by placing more counters in the second “part” section (2). Then move them all to the whole section to check that there are 7 all together.  Students are determining “What goes with 5 to make 7?” 5 + ___ = 7
• Here is an example of a result unknown subtraction story:  “Mom put 7 cookies on a plate. I ate 2 of them. How many cookies are still on the plate?” To do this start with the whole amount (7) in the large section. Then move the 2 that were eaten to a “part” section. Count how many are remaining in the “whole section” to find out how many are still on the plate?  7 – 2 = ____.
• How are number bonds related to fact families?  A fact family is one number bond shown with 2 addition and 2 subtraction statements.  Ex:  With number bonds 3 and 4 for the number 7, you can make 4 problems: 3 + 4 = 7;  4 + 3 = 7;  7-3 = 4;  and 7-4=3.

# Addition and Subtraction Part 1: Numerical Fluency

To be able to add and subtract, students normally pass through several phases as they build readiness for these operations with numbers.  As teachers, we know oral counting does not necessarily indicate an understanding of numbers and sets, just like reciting the alphabet doesn’t necessarily mean a child can recognize letters and sounds. Read ahead for freebies in the Part-Part-Whole section.

Numerical Fluency Continuum:  There are 7 steps to numerical fluency. If a child gets stuck on any of these steps, it may very likely halt their progress. Hopefully children move through these by the end of 2nd grade, but many students beyond that level have a breakdown which is likely because they missed one of these stages. Can you determine which of these stages your students are in?

1. One-to-one correspondence: The ability to count objects so each object counted is matched with one number word.
2. Inclusion of set: Does a child realize that the last number counted names the number of objects in the set? A child counts 5 objects.  When you ask how many, can they state “5.” If you mix them up after they just counted them, do they realize there are still 5?
3. Counting on: If a child counts 5 objects and the teacher then puts 2 more objects for the child to count, do they start all over or continue counting from 5?  5 . . . 6, 7.
4. Subitizing:Recognize an amount without physically counting (ie on dice, dot cards, fingers).
5. More Than / Less Than / Equal To:  Can a child look at two sets of objects and tell whether the second set is more, less, or equal to the first set. Can a child build a second set with one more, one less, or equal to the first set?
6. Part / Part / Whole: Compose and decompose sets by looking at the whole and the parts that make up the whole.
7. Unitizing: The child is able to move from counting by ones to count by sets / groups: fives, tens, etc.

# Daily Math Meeting Part 2: The basics and subitizing (KG-2nd)

In the next few posts, I will show various ways to conduct daily math meetings which you can incorporate into your daily schedule (as part of your normal morning meeting routine, or at the beginning of your daily math lesson). Daily Math Meetings (10-15 minutes) are vital for quickly reviewing math concepts and number sense in more visual and discussion based format. With primary students, this math meeting might center around the calendar bulletin board (or SMARTboard presentation). With intermediate students, it begins to take on the aspects of a “Number Talk” with a variety of computational strategies being the focus.

PreK – KG Level Components:

1. Counting
2. Subitizing
3. Days of the Week
4. Months of the Year
5. Graphing (weather, etc.)
6. Place Value (tens and ones: ten frames, straws, sticks, etc. to keep track of the days of school – working toward the 100th day)

1st – 2nd Grade Level Components:

1. The above plus . . .
2. Number Bonds (How can we break apart this number? Such as 10 = 3 + 7 or 6 + 4)
3. Place Value and skip counting using a 100 chart
4. Number of the Day (word form, base ten form, place on a numberline, tally marks, on a ten frame, expanded form, etc.)
5. Ordinal Numbers (using the calendar)
6. Counting money (add one cent each day and exchange pennies for nickels, nickels for dimes, etc.)

Subitizing

See my updated post on this topic by clicking here: http://cindyelkins.edublogs.org/2016/09/03/subitizing-what-does-that-mean/

This is such an important process in the continuum of counting, adding, and subtracting numbers. It means students can recognize certain quantities without physically counting each one. Continue reading

# Number Talks Part 1: Subitizing and Number Bonds KG-1st grade

A Number Talk is an opportunity to review number sense and operations by making it part of your daily math routine — so that what has previously been taught is not easily forgotten.

In this post I will expand on 2 methods for conducting a Number Talk session for KG-1st grade students (Subitizing and Number Bonds). Refer to a previous post (Sept. 10 – Daily Practice to Build Number Sense), in which I mentioned several other ways to review math concepts on a daily basis such as calendar topics, weather graphs, counting # of days of school, using a 100 chart, Choose 3 Ways, etc. Continue reading

Number Bonds are pairs of numbers that combine to total the target or focus number. When students learn number bonds they are applying the commutative, identity, and zero properties. PLUS, the information can be applied to both addition and subtraction problems. Number bonds of 10 are very critical to our place value system, and will enhance a student’s success with future addition and subtraction strategies such as use of an open number line.

• KG students should master number bonds to 5.
• First graders should master number bonds to 10.
• Second graders should master number bonds to 20.

# Daily Math Meeting Part 1: Ways to Build Number Sense K-5

To build number sense, students need frequent exposure or review of concepts you have previously introduced. There are many ways to build number sense on an on-going, informal basis – especially when you can squeeze in 10-15 minutes daily:

• During morning meeting time
• During a Number Talks session
• At the beginning of your math lesson
• At the end of your math lesson
• End of day closure time

I have included several of my power point slides on this topic as a PDF file (daily-practice-to-build-number-sense-pdf). Continue reading

# Subitizing – What does that mean?

by C. Elkins, OK Math and Reading Lady (updated post on 8-12-17)

The term “subitize” means to recognize quantity without counting. It is a concept recently added to the new OAS (Oklahoma Academic Standards). KG students should be able to “recognize without counting the quantity of a small group of objects in organized and random arrangements up to 10.” For first graders, the quantity is increased to 20 of “structured arrangements.” Subitizing is an important pre-requisite skill to learning addition and subtraction number combinations or number bonds.

Suggested items for the teacher to present this concept:

• Dot cards
• Ten frames and 2-color counters or tiles
• Dot dice
• Dominoes
• Tally marks