Math Problem Solving Part 4: Equal Groups

by C. Elkins, OK Math and Reading Lady

Are you ready for a way to help students solve multiplication and division word problems? I have developed a template that will help students record the given facts and think through the process involving equal groups. There are 3 basic types of equal groups problems:

  • # of groups and # in each group are known
  • # of groups and total are known
  • # in each group and total are known

Pictured here is an illustration of the strategy which can be used for multiplication and division problems. Get my FREE packet right here: Equal groups strategy with template.

One of the most important steps I recommend when working with equal groups problems is for students to brainstorm things that typically come in equal groups. There is a great book titled “What Comes in 2’s, 3’s, and 4’s” which can set the stage. It’s a picture book, but helpful to get kids’ brains warmed up.  Here are some visuals I made to illustrate the point. Click here for your FREE copy. They are included in this set of Equal groups pictures and list template

Kids may need your help to think of things to add to the list. See some hints below. This template and a full list is included in the above FREE download. 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 = _____

Continue reading

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

Math Meetings KG-3rd Grades

by C. Elkins, OK Math and Reading Lady

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:

1st Grade:

2nd Grade:

3rd Grade:

Back to School

by C. Elkins, OK Math and Reading Lady

Here are a few links to some previous posts regarding literacy and math you might be interested in to help you start your journey this year.  And in case you didn’t see it, I now have an easy link to most of my own free resources. Creating this was one of my summer projects. Click here to get it now, but it is also available in the black bar above. See info at the end for another $25 subscriber / comment challenge!!  Have a great start to your year and Enjoy!!!

  1. Getting to know you literature connection and math activity
  2. Building a classroom community (includes link to great team building practices)
  3. Writing part 1
  4. Guided Reading Part 1: Getting Started
  5. Guided Reading Part 2: Routines and Procedures
  6. Meaningful Student Engagement: Whole Class Reading
  7. Daily Math Meeting Part 1: Building Number Sense
  8. Daily Math Meeting Part 2: Subitizing
  9. Addition and Subtraction Part 1: Numerical Fluency
  10. Addition and Subtraction Part 3: Facts Strategies
  11. Multiplication Strategies Part 1
  12. Fractions Part 1: The basics

 

Math Art Part 2: Decomposing and composing squares and triangles

by C. Elkins, OK Math and Reading Lady

I wanted to show you another example of math art, this time using squares and triangles. This project also falls under the standards dealing with decomposing and composing shapes. With this project, students can create some unique designs while learning about squares, triangles, symmetry, fractions, and elements of art such as color and design. It would be a great project for first grade (using 2 squares) or for higher grades using 3 to 4 squares.

A great literature connection to this project is the book “The Greedy Triangle” by Marilyn Burns. (Click link to connect to Amazon.) The triangle in this book isn’t content with being 3-sided and transforms himself into other shapes (with the help of the Shapeshifter). Lots of great pictures showing real objects in the shape of triangles, squares, pentagons, hexagons, and more.

Marilyn Burns is a great math educator to check out, if you haven’t already. She has a company called Math Solutions (check out MathSolutions.com). Marilyn and her consultants have wonderful resources and advocate for constructivist views regarding math education. She is also the author of Number Talks and many math and literature lesson ideas.

The 4 Triangle Investigation

Materials needed:

  • Pre-cut squares 3″, 4″ or 5″ (I used brightly colored cardstock.)
  • Scissors and glue
  • Background paper to glue shapes to

Directions

  1. Model how to cut a square in half (diagonally) to make two right triangles. (I advocate folding it first so that the two resulting triangles are as equivalent as possible.)
  2. Guide students into showing different ways to put two triangles together to form another shape. Rule: Sides touching each other must be the same length. Let students practice making these shapes on their desk top (no gluing needed). 
  3. Help students realize they may need to use these actions:
    • Slide the shape into place
    • Flip it over to get a mirror image
    • Rotate it around in a circular motion to align the edges
  4. Students are then given 2 squares (to be cut into 4 triangles) and investigate different shapes they can make following the above rule. Here are some possibilities:
  5. As the teacher,  you can decide how many creations you want each student to attempt.
  6. These shapes can be glued onto construction paper (and cut out if desired).
  7. As an extension, shapes can be sorted according to various attributes:
    • # of sides
    • symmetry
    • # of angles
    • regular polygons vs. irregular

Continue reading

Math Art Part 1: Fraction circle art (3rd-5th)

by C. Elkins, OK Math and Reading Lady

Incorporating art with other subjects is a great way to engage students. In this post I will share one project which helps students gain hands-on experience with fractions. More to come in future posts.

Fraction Circle Art

This project is inspired by Ed Emberley’s book “Picture Pie.” This is my favorite of his collection in which he shows dozens of ways to use fractions of circles to create almost anything. This book features mostly animals, flowers, and geometric designs. Students start with a circle (pre-cut with a circle cutting press is best, but you can also make nice circles by tracing around a can or drinking glass and cutting them out). Then the circle is folded and cut into these different fractional parts to create the design: halves, fourths, eighths, and sixteenths. These pieces are manuevered (think translations!!) and combined to make the desired art.

The pictured creations were made by 3rd and 4th graders during a session I conducted with them at Eisenhower Elementary. Here are a few of them.  So nice!!

Continue reading

Fractions Part 6: Adding and Subtracting Fractions

by C. Elkins, OK Math and Reading Lady

Starting in 3rd grade, students start building understanding about adding and subtracting fractions by composing and decomposing simple fractions using concrete and visual models. Composing: 1/4 and 3/4 combine to make 4/4 (whole). Decomposing: 8/8 is made up of 2/8 and 6/8. In fourth grade, students begin to add and subtract fractions with like denominators, but should still be utilizing models, drawings, and number lines to illustrate and simplify. In fifth grade, students are expected to add and subtract fractions of all types (proper, improper, with unlike denominators, etc.).

With a firm foundation of composing and decomposing, partitioning, comparing, naming equivalent fractions, and understanding the relationship between certain fractions (such as halves / fourths / eighths / sixteenths; and thirds / ninths / sixths / twelfths; or fifths / tenths), then students are more prepared to perform operations with fractions. Here’s a great resource by Donna Boucher at Math Coach’s Corner: Composing and Decomposing Fractions activity on TPT ($6)

Estimating: This is an important part of operations with fractions. Do you expect your answer to be less than 1/2, more than 1/2, more than 1? How do you know? If I was adding 8/9 + 11/12, my answer should be about _____? It should be slightly less than 2 because both of these fractions are almost 1.

If I am adding 4/6 and 6/8, my answer should be more than 1 because each of these fractions are greater than 1/2.

Different strategies: There are many “tricks” or shortcuts available to show students how to quickly add, subtract, or multiply fractions. I believe these shortcuts are only useful after a students has a strong understanding of why and how to find a common denominator and equivalent fraction. These shortcuts do not help build conceptual understanding of fractions.  I will focus on ways to understand the why using visual and pictorial models. Get your FREE copy of the following guides by clicking HERE. Continue reading

Fractions Part 5: Equivalent Fractions

by C. Elkins, OK Math and Reading Lady

This is part 5 of a series of fractions posts. Thanks for sticking around! Through explorations with fraction manipulatives, pictures, and drawings, we hope students begin to notice there may be different ways to express the same area using fractional terms. To cut a sandwich into halves and eat one of the halves is the same as cutting the same sandwich into fourths and eating two of the fourths. Read on for several freebies about equivalent fractions.

Students can gain experience finding equivalent fractions using models in several ways: Fraction strips, area models, set models, bar / length models, and number lines. Then with a strong understanding using concrete and pictorial models, the student is ready to apply paper-pencil methods to name equivalent fractions. But remember to use the same size whole: As in this picture using pattern blocks, 2/3 of a trapezoid is NOT the same as 2/3 of a hexagon.

Fraction Strips: If you don’t have sets of fraction strips, here is a free resource Fraction, decimal and percent strips charts via Kim Tran (TPT).  Commercially available strips are also nice. But, with some 1″ strips of construction paper, students can create their own and probably learn a lot about the relationship between halves, fourths, and eighths as well as thirds, sixths, and twelfths in the process of partitioning and cutting them.

TIP:  Make sure students know how to read a fraction strip chart. A couple of years ago I was working with a third grade class and assumed they could readily see that 5/10 was equivalent to 1/2. But after confusing looks, I realized that I needed to physically show them how to follow a line vertically down the chart to find other fractions that were in line (by placing a ruler or long pencil along the vertical line). Another example.  “To find another fraction equivalent to 1/3, find the line at the end of the 1/3 section and trace it vertically down the page to see if there are other fractions that stop along that same line. You should see in the sixth’s line that 2/6 lines up, and in the ninth’s line that 3/9 lines up and in the twelfths line that 4/12 lines up.”

After cutting and labeling strips, then explore equivalent fractions (those with the same size length). Students should be able to generalize that different fractions can used to represent the same area.

Area models:  By covering or partitioning shapes, students should notice that even though the same area is covered or shaded,  the number and size of the parts can change. Continue reading

Fractions Part 4: Compare Fractions

by C. Elkins, OK Math and Reading Lady

Starting in 3rd grade, students start using words and symbols to read and write fractions (Oklahoma Academic Standards OAS 3.N.3.1), construct fractions (3.N.3.2), compose and decompose them (3.N.3.3), and order and compare them using models and number lines (3.N.3.4).  Fourth and Fifth graders continue to refine these skills. In this post, I will address different ways to compare fractions (keeping in mind the concrete-pictorial-abstract progression) by comparing numerators, comparing denominators, comparing to half, and utilizing knowledge of unit fractions. Students should have extensive experience utilizing models such as fraction strips, fraction circles, pattern blocks, number lines, pictures, and drawings to help build the concepts of fractional parts before being asked to put a <, >, or = sign between two fractions. See the end for a FREE comparing fractions guide.

In my opinion, determining if (or how) two fractions are equivalent is also a very important step when comparing fractions. However, regarding the OAS, students are not asked to represent or rename equivalent fractions until 4th grade (4.N.2.1). I will address equivalent fractions in the next post – just know that sometimes this skill goes hand in hand with comparing fractions. AND keep in mind that most of the standards for fractions through 4th grade stipulate “using concrete and pictorial models, fraction strips, number lines.” Students in 4th grade should not be expected to do abstract paper-pencil steps to simplify or “reduce” fractions to simplest terms, nor cross multiply to compare, etc. They need hands-on experience to more fully understand the concepts about fractions that are so difficult to grasp abstractly. Then in 5th grade students should have enough visual pictures in their head to solve operational problems with fractions. OK, that’s my soapbox. Don’t make it harder than it should be.

Materials to use: pattern blocks, fraction strips, fraction circles, cubes, tiles, two-color counters, Cuisenaire rods, number lines, paper plates, graham crackers, candy (m and m’s, skittles, etc.)

 

Ways to Compare (when using same size wholes – you can’t compare 3/4 of a donut with 1/2 of a birthday cake):

  • Using unit fractions:  If the fraction is a unit fraction, it has a 1 as a numerator. This should form the first type of comparison:  1/2 > 1/3 and 1/5 < 1/4 and 1/6 > 1/10, etc. This type of comparison is critical to fractional understanding.
  • Same denominator: When the denominators are the same, then compare the numerators. 2/4 > 1/4.
  • Same numerator:  When the numerators are the same, compare the denominators. For example: When comparing 2/5 with 2/10, since fifths are larger parts than tenths, 2/5 will be larger than 2/10. This is hard for some students to think about, because the smaller the number designated for the denominator, the larger the part (when comparing the same size whole). 
  • Unit fractions one away from the whole:  These are fractions in which there is one unit to be added to make it a whole (1). The numerators of these fractions will be one less than the denominator.  11/12 is 1/12 away from the whole (1). 7/8 is 1/8 away from the whole (1). Example: To compare 3/4 with 5/6, use manipulatives or a number line to see that 3/4 is 1/4 away from 1, while 5/6 is 1/6 away from 1.  Since 1/4 is a bigger part than 1/6, then 3/4 < 5/6.
  • Less than half? More than half?  Learn all of the fractions that equal half. While this might sound simple, students often have misconceptions that 1/2 is the only way to describe half, or that a 5 must be in the fraction to be half (because 5 is the midpoint when used on a number line for rounding). I ask students to recall their addition facts dealing with doubles from 2nd grade. Since 2 + 2 = 4, two is half of 4, and 2/4 = 1/2. Repeat that with other forms of 1/2. Students should learn that finding half of an even-numbered denominator should be figured quickly (7/14, 9/18, 25/50, 50/100, etc.). Then use knowledge of half to determine if a fraction is less than half or more than half. Since 7/14 = 1/2, then I know that 6/14 < 1/2 and 9/14 > 1/2.

Continue reading

Fractions Part 3: Misconceptions

by C. Elkins, OK Math and Reading Lady

The fractions focus today will be on some basic concepts that students should understand before they work to compare them, determine equivalent fractions, simplify them, use mixed fractions, or add / subtract them.  I am including a FREE copy of my Fraction Basics reference guide (click here), along with a photo of an anchor chart I made for a fourth grade class.

I have been rereading a book I love about fractions called “Beyond Pizzas and Pies, 1st Edition.” It has great examples of children’s misconceptions about fractions and lessons on how to try to remediate them. A recurring theme in the book is that while kids can learn “tricks” to help them solve fraction problems, they often do little to help students conceptualize what fractions are. Here’s a link to Math Solutions regarding this book: Beyond Pizzas and Pies (2nd Edition) Following are five  examples from the book that made an impact on me and my teaching (which I will go into more detail about on future posts). Continue reading

Fractions Part 2: Constructing and Drawing

by C. Elkins, OK Math and Reading Lady

The standards (CCSS or any state) use varied verbs to describe what students are to do regarding fractions: form, compose, construct, model, partition, draw, decompose, share, identify,  read, write, describe, order, and compare. Satisfying these standards can often be accomplished through use of concrete methods (manipulatives) and pictorial models (drawings). Remember the best understanding of concepts usually follows the concrete, pictorial, abstract progression (CPA). In other words, “Let’s make it, draw it, and then use numbers to represent it.”

Through constructing and drawing, students will  be prepared for further work with fractions, and they begin to conceptualize the relationship between the size of denominators, the numerators, and the whole. Click here for a FREE copy of the pictures you will see below (3 page pdf).

Form / Compose / Construct / Model:  Use smaller shapes to form or compose larger shapes (which is also a geometry std. in KG and 1st). Put together fraction pieces or puzzles to make the whole shape (circle, rectangle, hexagon, etc.). Use fraction pieces to demonstrate understanding by constructing models of area, set, and length.

These pictures show different ways students can use manipulatives to form, compose, construct, and model fractional parts (pattern blocks, fraction circles, tangrams, linking cubes, color tiles, fraction bars, Cuisenaire rods, two-color counters):

Partition / Draw / Decompose / Share: Split larger shapes into smaller fractional parts (halves, thirds, fourths, etc.). Divide (fair share) objects into equal groups. Use models to decompose a fraction in more than one way. Represent fractions on a number line.

I enjoy teaching children how to partition common shapes into fractional parts – because it involves drawing. Too often, if I just tell them to divide a rectangle or circle into fourths or sixths, I get something like this: Continue reading

Fractions Part I: Basics KG-2nd grade

by C. Elkins, OK Math and Reading Lady

This is the first post of several I will devote to fractions, starting with basic understanding in first grade and moving up toward operations with fractions in 5th and 6th grade. I would love to hear from you about your students successes and/or difficulties with fractions so I can be sure to address this topic to meet your needs. Free resources below.

What is a fraction?  A fraction represents a part of a whole. It consists of a numerator (which tells how many parts we are describing) and a denominator (how many parts the whole is divided into).

Some basics:

  • Fractional parts must be equal. This is a concept introduced in first grade. (See some lesson plan ideas below.)

    From Pinterest

  • The larger the denominator, the smaller the parts AND the smaller the denominator, the larger the parts — when comparing identical sized objects. You can’t compare 1/4 of a cookie with 1/4 of a cake. This is one of the hardest concepts to grasp – so lots of hands-on experience is needed.
  • Helpful manipulatives to use with fractions: pattern blocks, color tiles, Cuisenaire rods, fraction circles, fraction strips, fraction bars, graham crackers.
  • Be careful about always referring to fractions as “the shaded part.” While this might be true with pictures on worksheets, fractions can be described in these ways also: What fraction of the students are boys? What fraction of the pizza was eaten? What fraction of the candy bar is left? What fraction would belong here on the number line?
  • Lines do not necessarily define the fractional part. On the picture shown, the left shows 1/4 shaded. The right also shows 1/4 shaded, but students are likely to say 1/3. Why? Because they count the parts shown (3) and the shaded parts (1) and put that together as a fraction. A way to show this is still 1/4 is to show that the shaded part will fit into the whole shape 4 times.
  • A unit fraction: This is a fraction with 1 as the numerator (1/4, 1/8, etc.). It is one unit of the whole.
  • A fraction that is one unit away from a whole has a numerator one less than the denominator. Examples: 2/3, 3/4, 7/8, 11/12, etc. This is helpful to conceptualize when comparing fractions.
  • When reading a fraction number line, compare it to a bar model. Then it is easier to see it is the spaces are the focus, not the tick marks.

Continue reading

Making Sense of Division (3rd-5th)

by C. Elkins, OK Math and Reading Lady

Is division a dreaded topic on your list of objectives to teach? Like many math topics, students have a harder time understanding it most likely because it’s not something they use regularly in their lives. Students should understand why division is useful before they have to start solving division problems. In this post, I will focus on helping students see the relationship between subtraction, multiplication, and division both with concrete objects, pictures, and the partial quotients model. Freebies available below!!

Then let’s talk about what division really is — it is repeated subtraction; much the way multiplication is repeated addition. The issue is that repeated subtraction is not always very efficient. Here’s what I mean.

Let’s say I have the basic problem 25 ÷ 5.  I could start with 25 and then subtract 5, subtract another 5, another 5, another 5, and another 5 until I run out and reach zero.  I would have to do this 5 times. If I had 25 cookies that I wanted to share equally among 5 friends, I could do the “one for you, one for you, one for you, one for you, and one for you” process and still end up with 5 cookies for each. Or I could try “two for you, two for you,” etc. to make the action of passing out the cookies faster. When I get down to 5 cookies, I return to the “one for you . . .” to make it work.

With a larger problem such as 72 ÷ 6, I can again try subtracting 6 at a time until I reach zero. This would take 12 repetitions — not efficient, but still accurate. Could I subtract 12 at a time instead (2 groups of 6) to be more efficient? Or 18 at a time, or 24 at at time? This is the idea behind the partial quotients model I will refer to a little later. Continue reading

Volume: Concrete activities to increase understanding (Grades 3-6)

by C. Elkins, OK Math and Reading Lady

I have worked with several groups of 4th graders lately to build rectangular prisms as a way of learning more about volume. Typically students know the formula (length x width x height), but often lack the strategies or spatial ability to solve problems seen only in picture (2D form). Concrete (hands-on) experiences help cement knowledge when abstract formulas may pose difficulty. And . . . it’s always fun to “play” while doing math!!

Here are some observations regarding students’ difficulties:

  • Students often resort to counting the visible cubes, not realizing there are others on the back side – which can’t be seen on a 2D representation.
  • Students are unsure which dimensions are the length, width, and height.
  • Students lack multiplication skills.
  • Students don’t know the purpose of finding volume (other than counting the cubes).
  • Students are often confused when constructing prisms when one of the dimensions is 1. They weren’t sure this was even a possibility until they saw what it looked like (after building it!).

Some possible solutions:

Give students multiple opportunities to build 3D rectangular prisms:

  1. Length is the longest side on the base. Width is the shortest side on the base. The height is how tall it is.
  2. Use this variant of the L x W x H formula:  (Area of the base) x Height or (L x W) x H. With this mindset, the students need to find the length and width dimensions first. Finding the area of the base first helps them visualize the bottom layer. Then the height just means the number of total stacks or layers (with all of them matching the area of the base).
  3. Give students specific dimensions such as (5 x 3) x 2.
    • Using connecting cubes, build the base (bottom layer) first and determine the area (5 x 3 = 15).
    • Then build 1 more layer just like it so there is a total of 2 layers (the height).
    • Through this experience, students learn what a 5 by 3 base looks like . . . and that each layer of the height has the exact same area. It’s actually several layers stacked on top of each other.
    • To complete this prism, compute the area of the base (5 x 3) and then multiply it by the height (2). So (5 x 3) x 2 = 30 cubic units.
    • This experience shows why the measurement is stated as cubic units (because cubes were used).
    • Students may also see another way to solve the problem is to add the area of the base 2 times (15 + 15). Of course, multiplication is more efficient, but seeing the addition solution helps them realize each layer is the same.
    • Get this FREE Prism Building Activity and FREE Volume game for building rectangular prisms from me (click on links). For the game, you just need the recording sheet and 3 dice per pair of students.
    • Don’t have connecting cubes? Check with KG or 1st grade classes!

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.

Continue reading

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?”

Continue reading

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

by OK Math and Reading Lady

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.

Continue reading