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.
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:
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.
Try this from NMCT Illuminations sight (National Council for Teachers of Mathematics): Interactive ten frame
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)
4 is 1 away from 5. 4 is 2 and 2 (or like a square).
4 is 2 and 2. 4 is 3 and 1.
5 is 5 and 0. 5 is 4 and 1.
5 is 3 and 2.
Six is 5 and 1. Six is 3 and 3. Or the child may see the bottom as 4 and 2.
Six is 4 and 2. Six is 2 and 2 and 2
Learning station ideas for subitizing and number bonds with ten frames:
Match # cards to ten frames (use mini ten frames in resources above).
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.
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.
Put ten frame dot cards in order (least to greatest, or greatest to least).
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).
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?
This is Part 4 of a series about daily reading routines I recommend. Previously we have looked at read alouds, independent reading, and phonological awareness. Today’s focus is Phonemic Awareness. Some videos and freebies via TPT are linked below.
See link #3 below for FREE task cards from TPT
Phonemic Awarenessis under the umbrella of phonological awareness. This encompasses pre-reading skills associated with the sounds of language. Phonemic awareness is the part dealing with individual phonemes and how they can be identified, segmented, blended, and manipulated to create recognizable units or words . . . . the auditory portion. Students need a firm foundation with this aspect before they can adequately apply it to phonics and reading (which is where the visual aspects of the letters that make these sounds appears). So here are some basics about phonemic awareness:
Phonemes are the basic sound units. In the English language there are 44 of them (the consonants, the vowels, digraphs, etc.). Here is a good, short list from Orchestrating Success in Reading by Dawn Reithaug (2002).: 44 Phonemes However, if you want to go more in depth, then this link should satisfy your curiosity (or make you want to quit teaching spelling) from The Reading Well: 44 Phonemes in Detail
Onsets/rimes: The onset is the part of the word before the vowel. The rime is the part of the word including and after the vowel. Examples: In the word shop, /sh/ is the onset and /op/ is the rime. In the word bed, /b/ is the onset and /ed/ is the rime.
Identifying: When presented with a word orally, can a student identify the beginning sound or ending sound? Example: What is the beginning sound in the word moon? /m/. What is the last sound in the word jump? /p/. The brackets are used to represent the sound – the child is not asked to name the letter.
Segmenting: When presented with these words, can a student take the parts or individual sounds apart orally (segment)? Examples: bed = /b/ + /ed/ or /b/ + /e/ + /d/. Students would NOT be asked at this point to identify the letters that make those sounds, just the sounds.
Blending: When presented with these sounds, can a student put them together orally (blend) to form a word? Examples: /k/ + /at/ = cat; or /sh/ + /o/ + /p/ = shop
Manipulating: This involves adding, deleting, or substituting sounds. Example: What is /ap/ with /m/ added to the beginning? (map). What is /land/ without the /l/ sound? (and). Change the /b/ in bed to /r/. . . (red).
Daily teaching routine for Phonemic Awareness:
If using a reading series, check to see if there is a daily practice with words (like the examples above). Just a few minutes with the whole class is a good introduction and chance for you to observe / listen to who is or is not grasping these tasks.
Use simple pictures (such as fox): Ask students to do some of the following when you feel they are ready:
Name the picture and tell the onset and rime. /f/ + /ox/
Orally say all of the separate sounds /f/ + /o/ + /ks/. Use the length of your arm for these cvc words: tap shoulder and say /f/; tap inside elbow and say /o/; tap the wrist and say /ks/. Then run your hand along the whole arm to blend them back together.
Use an Elkonin sound box to show the distinct sounds. For fox, use a 3-part box. Push a chip into each box as each sound is being made (no letters yet, just chips, beans, cubes, pennies, etc.). Then blend all the sounds together. (I like to put an arrow at the bottom of the boxes and run my finger along it to remind students with a visual that the last step is to blend the sounds together.)
Change the /f/ to /b/. What word does that sound like? /b/ + /o/ + /ks/ = /box/
Change the /ks/ to /g/. What word does that sound like? /f/ + /o/ + /g/ = /fog/
Change the /o/ to /i/. What word does that sound like? /f/ + /i/ + /ks/ = /fix/
If you remove the /f/ sound, what is left? /oks/ or /ox/
Be sure to use short and long vowel words, digraphs, etc. because it’s all about hearing the separate parts – not about matching up the letters that make those sounds.
Follow up these same routines during guided reading and work station time. Here are 2 links from TPT (FREE) with some great sound box practice opportunities:
These routines will be very important once you feel they are ready to associate the letter(s) that make these sounds (via phonics, spelling, and writing). A phonics routine will be the next topic. So stay tuned!
Last time I focused on some basics about learning the number bonds (combinations) of 10 as well as adding 10 to any number. Today I want to show the benefits of making a 10 when adding numbers with sums greater than 10 (such as 8 + 5). Then I’ll show how to help students add up to apply that to addition and subtraction of larger numbers. I’ll model this using concrete and pictorial representations (which are both important before starting abstract forms).
Using a 10 Frame:
A ten frame is an excellent manipulative for students to experience ways to “Make a 10.” I am attaching a couple of videos I like to illustrate the point.
Model this process with your students using 2 ten frames.
Put 8 counters on one ten frame. (I love using 2-color counters.)
Put 5 counters (in another color) on the second ten frame.
Determine how many counters to move from one ten frame to the other to “make a 10.” In this example, I moved 2 to join the 8 to make a 10. That left 3 on the second ten frame. 10 + 3 = 13 (and 8 + 5 = 13).
The example below shows the same problem, but this time move 5 from the first ten frame to the second ten frame to “make a 10.” That left 3 on the first ten frame. 3 + 10 = 13 (and 8 + 5 = 13). Continue reading →
I’m sure everyone would agree that learning the addition / subtraction facts associated with the number 10 are very important. Or maybe you are thinking, aren’t they all important? Why single out 10? My feelings are that of all the basic facts, being fluent with 10 and the combinations that make 10 enable the user to apply more mental math strategies, especially when adding and subtracting larger numbers. Here are a few of my favorite activities to promote ten-ness! Check out the card trick videos below – great way to get kids attention, practice math, and give them something to practice at home. Continue reading →
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?
One-to-one correspondence: The ability to count objects so each object counted is matched with one number word.
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?
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.
Subitizing:Recognize an amount without physically counting (ie on dice, dot cards, fingers).
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?
Part / Part / Whole: Compose and decompose sets by looking at the whole and the parts that make up the whole.
7 is the “focus number”
1 and 6 are bonds of 7
Unitizing: The child is able to move from counting by ones to count by sets / groups: fives, tens, etc.
This is the Part 3 of Number Talks. If you are just tuning in, please refer to NT Parts 1 and 2. As I mentioned before, conducting a Number Talk session with your students is a chance for them to explain different ways to solve the same problem. This is meant to highlight strategies which have already been taught.
Click below to watch 2 videos of how to conduct a Number Talk session with intermediate students. You will see many strategies being used.
Addition and Subtraction Strategies: I like using the methods listed below before teaching the standard algorithm. This is because they build on a solid knowledge of place value (and number bonds 1-10). If your students are adding and subtracting using the standard algorithm and can’t adequately explain the meaning of the regrouping process in terms of place value, then try one of the following methods. In many cases, I will ask a student the meaning of the “1” that has been “carried” over in double-digit addition. About 85% of the time, the student cannot explain that the “1” represents a group of 10. When adding the tens’ column, they often forget they are adding groups of 10 and not single digits. So they get caught up in the steps and don’t always think about the magnitude of the number (which is part of number sense). You will notice teachers write the problems horizontally in order to elicit the most strategies possible.
Place Value Decomposition
Open Number Line (to add or subtract)
Here are some possible Number Talk problems and solutions:
Notice use of place value and number bonds.
These methds build strong number sense.
I call this “Facts Of.” You can use any number.
Multiplication and Division Strategies: I like using these methods before teaching the standard algorithms. Again, they build a solid understanding of place value, the use of the distributive property, and how knowledge of doubling and halving increases the ability to compute problems mentally. Once these methods have been learned, then it is easy to explain the steps in the standard algorithm.
Doubling and Halving
Here are some possible Number Talk problems and solutions:
Use the known (6 x 2) to learn the unknown problem.
Use the distributive property!
To divide by 4, halve the number twice. To divide by 8, halve the number 3 times.
For 1st -3rd grade students: Refer to “Number Talks Part I” (posted Nov. 12, 2016) for ways to conduct a Number Talk with KG and early 1st grade students (focusing on subitizing and number bonds). For students in 1st – 3rd grade, place extra emphasis on number bonds of 10.
Problem: 8 + 6
Problem: 15 + 9
Problem: 18 – 9
Problem: 35 + 22
Problem: 99 + 6
Problem: 200 – 48
Write a problem on the board, easel, or chart tablet with students sitting nearby to allow for focused discussion. Have the following available for reference and support: ten frame, part-part-whole template, base ten manipulatives, and a 0-100 chart. Present addition and subtraction problems to assist with recall of the following strategies. If time allows, post another similar problem so students can relate previous strategy to new problem. Students show thumbs up when they have an answer in mind. The teacher checks with a few on their answer. Then he/she asks, “How did you solve this problem?” The teacher writes how each student solved the problem.
Reference these videos on youtube.com for examples by clicking on the link:
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 →