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 have 1 penny in one pocket. I have 6 more pennies in another pocket. This is a Join or “some and some more” story structure.
Many teachers I work with have asked for advice on problem solving in math. Students often don’t know how to approach them or know what operation to use. Should teachers help students focus on key words or not? What about the strategies such as CUBES, draw pictures, make a list, guess and check, work backwards, find a pattern?
While all of those strategies definitely have their purpose, I find we often give kids so many steps to follow (underline this, circle this, highlight that, etc.) that they lose sight of what the problem is basically about.
In this post, I will focus on two basic questions (who and what) and a simple graphic organizer that will help students think about (and visualize) the actions in a one-step Join story problem. KG and first grade students can act out these actions using story mats or ten frames. Late first through 5th grade can use a part-part-whole box. There are two FREE items offered.
These are the types of problems I will focus on in the next few posts.
Join (also referred to as SSM – Some and Some More)
Separate (also referred to as SSWA – Some, Some Went Away)
JOIN problems have 3 versions:
a + b = ___ (The result is unknown.)
a + ____ = c (How the story changed is unknown / missing addend.)
____ + b = c (The start is unknown / missing addend.)
They can also be referred to as “Some and Some More” stories (SSM). This means, you have some and you get some more for a total amount. These present themselves as additive stories, but it doesn’t necessarily mean you have to add to solve them. The second and third version above are often referred to as missing addend problems. Continue reading →
This week I will focus on subtraction problem structures. There are two types: separate and compare. I suggest teaching these models separately. Also, some part-part-whole problems can be solved using subtraction. I will refer to the same terms as in addition: start, change, result. You can also use the same materials used with addition problems: part-part-whole templates, bar models, ten frames, two-color counters, number lines, and connecting cubes.
The goal is for students to see that subtraction has different models (separate vs. comparison) and an inverse relationship with addition — we can compose as well as decompose those numbers. Knowledge of number bonds will support the addition / subtraction relationship. Here is the same freebie I offered last week you can download for your math files: Addition and Subtraction Story Structure Information The six color anchor charts shown below are also attached here free for your use: Subtraction structure anchor charts
Separate: Result Unknown
Example: 10 – 4 = ____; There were 10 cookies on the plate. Dad ate 4 of them. How many are left on the plate?
Explanation: The problem starts with 10. It changes when 4 of the cookies are eaten. The result in this problem is the answer to the question (how many are left on the plate).
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, it is 10.) Do we know what changed? (Yes, 4 cookies were eaten so we take those away.) How many cookies are left on the plate now? (Result is 6.)
Reinforce the number bonds of 10: What goes with 4 to make 10? (6)
Draw a picture to show the starting amount. Cross out the items to symbolize removal.
Show the problem in this order also: ____ = 10 – 4. Remember the equal sign means the same as — what is on the left matches the amount on the right of the equal sign.
Ask your students this question: “What does the equal sign mean?” I venture many or most of them will say, “It’s where the answer goes,” or something to that effect. You will see them apply this understanding with the type of problem in which the answer blank comes after the equal sign such as in 5 + 6 = _____ or 14 – 9 = _____. This is the most common type of problem structure called Result Unknown. I was guilty of providing this type of equation structure a majority of the time because I didn’t know there actually were other structures, and I didn’t realize the importance of teaching students the other structures. So . . . here we go. (Be sure to look for freebies ahead.)
Teach your students the equal sign means “the same as.” Think of addition and subtraction problems as a balance scale: what’s on one side must be the same as the other side — to balance it evenly.
Let’s look at the equal sign’s role in solving the different addition / subtraction structures. I will be using the words start to represent how the problem starts, the word change to represent what happens to the starting amount (either added to or subtracted), and the result. In this post I will be addressing the join / addition models of Result Unknown, Change Unknown, Start Unknown and Part-Part-Whole. I will highlight the subtraction models (Separate and Comparison) next time. Knowing these types of structures strengthens the relationship between addition and subtraction.
Three of these addition models use the term join because that is the action taken. There are some, then some more are joined together in an additive model. One addition model, however, is the part-part-whole model. In this type there is no joining involved – only showing that part of our objects have this attribute and the other part have a different attribute (such as color, type, size, opposites).
Example: 5 + 4 = ____; A boy had 5 marbles and his friend gave him 4 more marbles. How many marbles does the boy have now?
Explanation: The boy started with 5 marbles. There was a change in the story because he got 4 more. The result in this problem is the action of adding the two together.
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, it is 5.) Do we know what changed? (Yes, his friend gave him more, so we add 4.) What happens when we put these together?(Result is 9.)
Show the problem in this order also (with result blank first instead of last) : ____ = 5 + 4
Common questions: How many now? How many in all? How many all together? What is the sum?
Join: Change Unknown
Example: 5 + ____ = 9; A boy had 5 marbles. His friend gave him some more. Now he has 9 marbles. How many marbles did his friend give him? You could also call this a missing addend structure.
Explanation: This problem starts with 5. There is a change of getting some more marbles, but we don’t know how many yet. The result after the change is (9). It is very likely students will solve like this: 5 + 14 = 9. Why? Because they see the equal sign and a plus sign and perform that operation with the 2 numbers showing. Their answer of 14 is put in the blank because of their misunderstanding of the meaning of the equal sign.
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, he started with 5 marbles.) Do we know what changed in the story? (Yes, his friend gave him some marbles, but we don’t know how many.) Do we know the result? (Yes, he ended up with 9 marbles.) So the mystery is “How many marbles did his friend give him?”
Count up from the start amount to the total amount. This will give you the change involved in the story.
Reinforce number bonds by asking, “What goes with 5 to make 9?” or “What is the difference between 5 and 9?”
Join: Start Unknown
Example: ____ + 4 = 9; A boy had some marbles. His friend gave him 4 more. Now he has 9. How many did he have to start with?
Explanation: The boy had some marbles to start with, but the story does not tell us. Then there is a change in the story when his friend gives him some more (4). The result is that he has 9.
Teaching and practice suggestions:
Ask questions such as: Do we know how many marbles the boy started with? (No, the story didn’t tell us.) Do we know what changed in the story? (Yes, his friend have him 4 marbles.) Do we know the result? (Yes, he had 9 marbles at the end.)
Count up from 4 to 9.
Reinforce knowledge of number bonds by asking, “What goes with 4 to make 9?”
This is part three in a series of strategies regarding addition and subtraction strategies. This part will focus on a variety of strategies to help toward memorization of facts, meaning automatic computation. While children are learning their number bonds (building up to 5 in KG, to 10 in first grade, and to 20 in second grade), there are other facts which cross several number bonds that students can work towards. These strategies to build mental math automaticity are highlighted below. Get some freebies in the section on doubles / near doubles.
Identity (or Zero) Property:
The value of the number does not change when zero is added or subtracted.
3 + 0 = 3
9 – 0 = 9
The answer is always zero when you take away / subtract all.
9 – 9 = 0
50 – 50 = 0
Adding 1 or Subtracting 1:
Adding 1 results in the next number in the counting sequence.
Subtracting 1 means naming the number that comes right before it in the counting sequence.
With manipulatives, lay out an amount for student to count. Slide one more and see if he/she can name the amount without recounting.
Do the same as above, but take one away from the group to see if he/she can name the amount without recounting.
Show this concept using a number line.
6 + 1 = 7; 26 + 1 = 27
7 – 1 = 6; 37 – 1 = 36
After +1 or -1 strategies are in place, then go for +2 or -2 for automatic processing.
Next-Door Neighbor Numbers:
If subtracting two sequential numbers (ie 7 subtract 6), the answer is always one because you are taking away almost all of the original amount.
Help students identify these types of problems: 8-7; 10-9; 98-97; 158-157
Guide students to writing these types of problems.
Relate these to subtracting 1 problems. If 10-1 = 9; then 10 – 9 = 1.
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.
7 is the “focus number”
1 and 6 are bonds of 7
Sample recording page
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.
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:
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.
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: