# 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.

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

# 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.

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

# Graphic Organizers for Math

by C. Elkins, OK Math and Reading Lady

Here are some cool graphic organizers for your math files!  Make sets of them, laminate or put in plastic sleeves, and use them over and over again!  Graphic organizers help students stay organized and teach them how to complete problems neatly. They are also a great way for students to show different strategies for the same problem. While primary students may need an already-made graphic organizer, intermediate students should be taught how to duplicate them on their own to use whenever the need arises – so the simpler, the better! With repeated use, students are more likely to utilize them regularly in their daily work (and on their scratch paper with assessments).

This one has ten frames and part-part-whole models. In my opinion, these are essential when working with K-2 students because they help children with subitizing, number bonds, and addition / subtraction facts.  If you are using Saxon, you are missing these important strategies!!:

Here’s one to show fractions (area, set, length models)

Need a template for students to make arrays? This one is ready!  I love showing students how to break an array into smaller parts to see how multiplication (or division) facts can be decomposed.  Example:  Make a 6 x 7 array.  Section off a 6 x 5 part. Then you have a 6 x 2 part left over.  This proves:  6 x 7 = (6 x 5) + (6 x 2).  Or — 6 x 7 = 30 + 12 = 42

This graphic organizer shows 5 different multiplication strategies using 2 digit numbers, and a blank one for students to record their thinking. Very handy!!  One of my favorite strategies is partial products. I highly recommend this one before going to the std. algorithm because students decompose the problem by place value and must think about the whole number and not just the parts.

Do your students need something to help them see the different models for a decimal? Try out this graphic organizer. Students will utilize the pictorial forms as well as the abstract.

Do your students know that .7 (or 7/10) is the same as .70 (or 70/100)?  Using this dual set of tenths and hundredths grids will help them see why this is true!

Be sure to check out my FREE templates and organizers (see black bar above “links . . .”)  Please share your favorite graphic organizers for math!  Enjoy!!

# All About 10: “Make a 10” and “Adding Up”

by C. Elkins, OK Math and Reading Lady

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.

Let’s say the task is to add 8 + 5:

• 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

# All About 10: Fluency with addition and subtraction facts

by C. Elkins, OK Math and Reading Lady

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

# Math Problem Solving Part 3: Comparing problems

by C. Elkins, OK Math and Reading Lady

Solving comparison problems involves a different thinking process than addition or subtraction problems. In my previous posts Math Problem Solving Parts 1 and 2, I focused on addition and subtraction problems in which I advised students to think of the verbs used (found, bought, earned, spent, lost, gave away, etc.) to help visualize what is going on in the story.  In comparison problems, there are no additive or subtractive processes present – just analyzing which has more, which has less, and how much more or less one amount is compared to another.

I love using a double bar method for comparison problems. It can be represented with manipulatives for a concrete experience (ex: connecting cubes, tiles) or with a drawing of two rectangular bars for a pictorial representation. These two visual models provide students with the concept of comparison — they can see right away which has more, which has less. Then the information can be used to compare the two amounts. Look for the FREE resource at the end of this post.

With manipulatives:

I recommend the use of manipulatives when dealing with students’ first experience with comparison problems. Dealing with quantities less than 20 make using manipulatives manageable. I will show this in a horizontal format, but it can certainly be done vertically. You can also apply this quite well to graphing problems.

Problem:  I have 12 crayons. My friend has 8 crayons. How many more crayons do I have than my friend?

1. Determine that this problem is not an addition / subtraction process (because no one is adding to what they have and no one is giving away what they have), but in fact a comparison problem.
2. Be cautious about focusing on the question”how many more” and telling students that when they see this they need to subtract. When we tell kids to focus on the specific words in a question with a “rule” for adding or subtracting, they start to lose sight of the actual story.
• Consider that this question can also be used with a SSM (some and some more) story which is not a comparison problem:  What if it read: “I have 8 nickels. I want to buy a candy bar that costs 80 cents. How much more money do I need?” This story means: I have some (40 cents) and I need some more (?) so that I have a total of 80 cents. This is a missing addend or change unknown story: 40 + ___ = 80.  Even though it could be solved in a similar manner as a comparison story, we want students to be able to tell the difference in the types of stories they are solving.
3. Determine who has more (represented by yellow tiles), who has less (green tiles).
4. The crayon problem can be solved by lining up 2 rows of manipulatives (pictured):
• Notice the extras from the longer bar. Count them (4)., or
• Count up from 8 to 12 to find the difference.
• Even if the question was “How many fewer crayons does my friend have?” it would be solved the same way.

With pictorial double bars:

Problem Type 1 (Both totals known):  Team A scored 85 points. Team B scored 68 points. How many more points did Team A score than Team B?

1. Determine this problem is not an addition / subtraction process (because the teams are not gaining or losing points).
2. Ask “Who?” and “What?” this story is about:  Team A and B and their scores.
3. Draw double bars (one longer, one shorter) which line up together on the left side.
4. Label each bar (Team A, Team B).
5. For the team with the larger amount (Team A), place the total outside the bar (85).
6. For the team with the smaller amount (Team B), place the total inside the bar (68).
7. Make a dotted line which extends from the end of the shorter bar upward into the longer bar.
8. Put a ? inside the extended part of the longer bar. This is what you are trying to find.
9. To solve, there are 2 choices:
• 68 + ____ = 85     This choice might be preferred for those with experience using mental math or open number lines to count up.
• 85 – 68 = _____

# Math Problem Solving Part 2: Separate (aka Some, Some Went Away)

by C. Elkins, OK Math and Reading Lady

In the previous post, I addressed problems dealing with an additive process (join; aka SSM).  In this post, I will show you some models to use for these types of problems:  Separate; aka Some, Some Went Away — SSWA.  I will share some models which are great for KG stories as well as templates that are helpful for intermediate students to use, especially when dealing with missing addend types.

As I previously mentioned, it is my belief that students should focus more on the verb / action in the story and not so much with the key words we often tell kids to pay attention to. Brainstorm actions that signify a subtractive process.  Post it in the class.  Keep adding to it as more actions are discovered. With this subtractive action, kids should quickly realize that there should be less than we started with when we take something away. Here is a FREE poster showing some of the most common subtraction action verbs. Click HERE to get your copy.

Some of the work mats pictures below come from the following source. These are great for KG-2nd subtraction storytelling.   Subtraction Pack: A Pinch of Kinder by Yukari Naka

Like with all story problems, I model how I reread the problem several times.

• First read — Just read it
• Second read –Identify who and what the story is about (the action).
• Third read — Decide what to do with the numbers. Is a given number the wholetotal amount or part of the amount? Do I know how the story started? How it changed? The result?

Here are 3 types of subtraction story structures: Continue reading

# Math Problem Solving Part 1: Join (aka Some and Some More)

by C. Elkins, OK Math and Reading Lady

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.

1. Join (also referred to as SSM – Some and Some More)
2. Separate (also referred to as SSWA – Some, Some Went Away)
3. Part-Part-Whole
4. Comparing
5. Equal groups

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

# Addition and Subtraction Part 5: Separate and Comparison Problem Solving Structures KG-4th

by C. Elkins, OK Math and Reading Lady

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.

# Addition and Subtraction Part 4: The Equal Sign and Join Problem Solving Structures KG-4th

by C. Elkins, OK Math and Reading Lady

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).

Helpful materials to teach and practice these strategies are bar models, part whole templates, a balance scale, and ten frames using cubes and/or two color counters. Here are 2 free PDF attachments. First one: Addition and Subtraction Story Structure Information. Second one is a copy of each of the 4 anchor charts shown below:  Join and part-part-whole story structure anchor charts

Join: Result Unknown

• 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?”

# Addition and Subtraction Part 3: Facts Strategies KG-3rd

by C. Elkins, OK Math and Reading Lady

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

Subtracting All:

• 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.
• Show on a number line.

Doubles (with freebies): Continue reading

# 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

by C. Elkins, OK Math and Reading Lady

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.

# 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.

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. Continue reading

# Daily Math Meeting Part 5: Using the 100 Chart and “Guess My Number”

by C. Elkins, OK Math and Reading Lady

This post will focus on ways to use a 100 chart to teach or review several math standards in the number sense and number operations strands (all grade levels). Each of these strategies can be completed in just a few minutes, making them perfect for your daily math meeting. Choose from counting, number recognition, number order, less/greater than, odd/even, addition, subtraction, multiplication, number patterns, skip counting, mental math, 1 more/less, 10 more/less, etc.

You can use a 1-100 chart poster on the smartboard, in poster form, or as a pocket chart. The pocket chart is the most versatile. See an example here: enasco.com pocket chart   Here is also a link to little colored transparent pieces that can be placed in the pockets to highlight chosen numbers: enasco.com pocket chart transparent inserts   I often show students that a 100 chart is actually just a giant number line all squished together instead of spread out across the room. To do this, I print off a chart, cut it into rows, tape the rows together, then highlight each multiple of 10. Second concept is that the lower numbers are at the top, and the higher numbers are at the bottom.

Counting, Number Order, and Place Value

• Instead of starting with a full 100 chart, start with an empty chart. Add 1 number per day in order, building toward the 100th day of school. This would be suggested for KG level.
• For other grade levels: Start with the numbers 1-10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. Put the rest of the number pieces in a jar, baggy, or container. Draw one or more numbers at random each day and assist students in placing the number where it belongs. Example:  If you draw out 45, let’s look at the one’s place (5) and know that it belongs in the same column as the 5. Let’s look at the ten’s place. We know it is greater than 40, but less than 50 so this helps us know which row it belongs in. As you progress, start using the currently placed numbers to help locate the new numbers. “I need to place 67.  I see 57 is already on our chart and know that 67 is ten more, so I place it directly underneath.”
• Number Thief Game:  After your chart is filled, try this game. After the children have left for the day, remove a few of the pieces. Then during your math meeting the next day, the children try to identify the missing numbers. Read how this blogger describes it:  “Swiper” at petersons-pad.blogspot.com
• Number locating: Just practice locating numbers quickly. If asked to find 62, does the student start at 1 and look and look until they find it? Or can they go right to the 60s row?
• Place Value Pictures:  You can’t do this on your hundred chart at meeting time, but there are dozens of picture-making worksheets available for free on TPT in which students follow coloring directions to reveal a hidden picture. Students get much better with locating numbers quickly with this type of practice.

Guess My Number: This is great for reviewing various number concepts. Here are a variations of guessing games. You can use with 1-100 chart, or 100-200, etc.

1. Teacher writes a number secretly on a piece of paper (ex: 84). The teacher gives a single clue about the number, such as: “My number is greater than 50.” Then let 2-3 students guess the number. Confirm that they at least guessed a number greater than 50. Redirect if not. If you have the little colored inserts, place one in each of the incorrect numbers so students will know what was already guessed. If you don’t have those, just write the guessed numbers somewhere where students can see.  Give a new clue after every 2-3 guesses until someone guesses the number.  After guessing correctly, I always show the students the number I had originally written down so they will know I was on-the-level. Here are some example clues for the secret number 84: My number is even.  In my number, the one’s place is less than the ten’s place.  My number is less than 90. My number is greater than 70. If you add the 2 digits together, you get 12.  The one’s digit is half of the ten’s digit. Again, affirm good guesses because at first there may be several numbers that fit your clue.

# Daily Math Meeting Part 4: Number of the Day/Week and Fun Facts

by C. Elkins, OK Math and Reading Lady

This is part 3 of my “Daily Math Meeting” posts. I will share several different fun and motivational math activities that can be done in just a few minutes on a daily basis — all of them building number sense and reviewing concepts of subitizing, number bonds, addition, subtraction, less, greater, even, odd, etc.

Number of the Day / Week

You can look on Pinterest or TPT and see many good resources on this topic – from daily review sheets to bulletin board products. Here’s my take on it (depending on your grade level).  If you are KG, then I suggest a number of the week, building from 1-10 at first (for the first 10 weeks). Focus on #1 the first week, #2 the second and so on. Really go in depth with each number, revealing a little bit each day. Then after the 10th week, repeat. This will give students adequate time to focus on each number in depth. See the attached PDF for some of my slides regarding this topic. daily-practice-to-build-number-sense-pdf

Monday:  “Our number this week is one.” Here’s what it looks like (show the numeral 1).” Students say the number and make it in the air. Teacher shows how to write it. Then show a representation of the number (such as putting something in a jar or posting on the board).

Tuesday-Thursday: Review the above and then show another way to represent the number (maybe 1-2 more each day). Examples:  Five or Ten frame, dice, domino, fingers on a hand, place on the number line, word form, tally mark, random dot. Talk briefly about how the patterns help you remember the amount without counting them (which is subitizing). When showing the 4 on a dice, notice that “if you connect the corners, you make a square.” Then when showing 5, notice that, “it’s like 4, but with a dot in the middle.”

Friday: Quickly review previously posted information about your number. Share a problem involving the number.  “I had nothing in this jar, and then I put 1 marble in it. How many marbles are in there now?” Along with this type: “Look, I have a marble in my jar. That means I have how many? (Students answer with “one.”). “What if I take this 1 marble out? How many will there be in the jar?” Share other concepts of this number such as (uno, single for one; or double, twin, duet for two, etc.)

When working with numbers 2-10: You will also start focusing on number bonds. Using 2-color counters on a ten frame, show (and let students think of) different ways to make the number of the week. Example for #5: 1 red, 4 yellow; 2 red, 2 yellow; 3 red, 2 yellow; 4 red, 1 yellow; 5 red, 0 yellow; 0 red, 5 yellow. You don’t even need to make an equation yet. Just say “1 and 4 makes 5; 2 and 3 makes 5 . . .”

For first or second grade: I have two thoughts on this. You could do a number of the day utilizing the calendar date as your number. This means if it’s the 14th of the month, you are focusing on #14. This also means you would repeat these numbers each month – thus giving more exposure to the numbers students are most likely using on a regular basis. You could add the following concepts to your discussion: place value with tens/ones (in straw bundles, stick bundles, or posting sticky dots on ten frames); expanded notation (14 = 10 + 4); concepts of odd and even, and how to make the number using coins.

Second thought is this:  Keep track of the number of days of school (for those of you who like to celebrate the 50th and/or 100th day of school), but choose a number of the day or week to focus on so you can review those very important number concepts and number bonds with numbers from 0-20. Part of your board could have a whole/part/part section to show a way to break apart your number. Continue reading

# Number Talks Part 3: Computational Strategies 3rd-5th grades

by Cindy Elkins, OK Math and Reading Lady

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.

Number Talk 3rd grade 90-59 = ____

Number Talk 5th grade 12 x 15 = ___

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.

• Partial Sums
• Place Value Decomposition
• Expanded Notation
• Compensation
• Open Number Line (to add or subtract)

Here are some possible Number Talk problems and solutions:

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.

• Area Model
• Partial Products
• Distributive Property
• Doubling and Halving
• Partial Quotients

Here are some possible Number Talk problems and solutions:

Enjoy your Number Talks!!

# Number Talks Part 2: Strategies and decomposing with 1st-3rd grade

by Cindy Elkins, OK Math and Reading Lady

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.

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.