Multiplication Concepts Part 3: Equal Groups

by C. Elkins, OK Math and Reading Lady

Thanks for checking in on part 3 of my multiplication posts. Focus will be on the equal groups strategy — looking at how students can efficiently use this strategy to help learn basic multiplication facts. My angle will be at the conceptual level by using concrete and pictorial methods.

Basics:

  • Instead of in array or area format, equal groups are separate groups.
  • The “x” means “groups of.”  So 3 x 4 means “3 groups of 4.”

What things normally come in equal groups? Conduct a brainstorming session. I love the book “What Comes in 2’s, 3’s, and 4’s” as a springboard. After reading the book, let students brainstorm other things that come in equal groups. See the pictures below for some more ideas. After some internet research, I also made this attached list to use (in case you or your students draw a blank): click here: Equal groups pictures and list template

Use these lists to help students generate stories about equal groups. When students can create (and maybe illustrate) their own stories, they are much better at solving problems they must read on their own. This also helps students think carefully about what in the story constitutes a “group” and what the “groups of” represents:  

  1. There were 5 bowling balls on the rack. If you count all of the holes (3 per ball), how many holes are there all together? (5 x 3). The bowling balls are the groups. The holes are what is being counted in each group.
  2. How many numbers are shown on 3 clocks? (3 x 12). The clocks are the groups. The numbers are what is being counted in each group.
  3. I bought 8 pair of earrings. How many earrings are there? (8 x 2). The pairs are the groups.
  4. Seven ladybugs were crawling on the leaves. How many legs would there be? (7 x 6). The ladybugs are the groups. The legs are what is being counted in each group.

Ways to show equal groups with objects and drawings:

  • Hula hoops (great to use these in PE class to emphasize multiplication)
  • Embroidery hoops
  • Circles of yarn
  • Dishes:  cup, bowl, plate, tray
  • Baskets
  • Shelves

Objects to use to show equal groups:

  • people
  • cubes
  • tiles
  • mini erasers
  • teddy bear manipulatives
  • base ten materials
  • food: pinto beans, macaroni, cereal, candy
  • practically anything you have an abundance of!!

Teaching concepts regarding equal groups:

  • When students are placing objects or drawing inside, do they randomly place objects? Or do they organize them to enable ease in counting? Showing students how to organize the objects in each set contributes to their knowledge of equal groups — AND it’s a big help to you as the teacher as you check on students. If the dots are randomly placed, the teacher and student must count one at a time to check. If they are organized, teacher and student can tell at a glance if the amount in each group is correct. Notice the difference below: Which ones show a student’s understanding of 9? Which ones can a student or teacher check rapidly?

  • When counting the objects or drawings to determine the product of these equal groups, are students counting one at a time? Or are they counting in equal groups (such as by 2’s, 5’s, 3’s, etc.)? If we allow students to just count by ones, then they are not practicing multiplication, just counting!!

Activities to practice equal groups strategy:

  1. Circles and Stars:  Roll a dice once. This is the number of circles to draw. Roll a dice again. This is the number of stars to draw inside. If played with a partner, students can keep track of their totals to determine a winner. Dice can be varied depending on the facts that need to be practiced. A spinner can also be used. (See picture at beginning of this post.)
  2. Variation of above:  Use other materials (such as those listed above).
    • Dice roll #1 = # of cups. Dice roll #2 = number of cubes
    • Dice roll #1 = # of hoops. Dice roll #2 = # of pinto beans
    • Dice roll #1 = # of plates. Dice roll #2 = # of Cheerios
  3. Write and illustrate stories:  Provide a problem for students to illustrate (example:  6 x 3 or 3 x 6).  Then each student can decide how to form the story and illustrate. I always tell students to choose items they like to draw to make their story. Here are some examples.  See some examples from former students.
    • There were 6 monsters in the cave.  Each monster had 3 eyeballs. How many eyeballs all together?
    • Six princesses lived in the castle. They each had 3 ponies. How many ponies in all?
    • There are 3 plants in the garden. They each have 6 flowers. How many flowers are in my garden?
    • I made 3 pizzas. Each pizza had 6 slices. How many slices of pizza did I make?
  4. PE Class activities:  If your PE teacher likes to help you with your learning objectives, let them know you are working on equal groups strategies. While I’ve not done this personally, I think having relay races related to this would work perfectly. For example, the teacher presents a problem and each team must use hula hoops and objects to show the problem (and the answer).
  5. Try these story books about multiplication:
  6. Equal groups story problems to solve:  See my previous post related to this. You will find some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click HERE

Enjoy!!  

 

Division Basics Part 2: Equal Groups

by C. Elkins, OK Math and Reading Lady

Last post featured division using arrays and the area model.  This post will focus on helping children see division as equal groups. Most of us have used the “plates of cookies” analogy to help kids see how to represent equal groups in a drawing.  I will just take that a few more steps to increase efficiency.

Much like multiplication, there are different aspects of division children should get familiar with:

  • Arrays 
  • Equal Groups
  • Repeated Subtraction
  • Number lines
  • Skip counting

In this post, I will break down the benefits of equal groups models to help children understand division (and how it is related to multiplication). Check out the freebies within this post.

If you haven’t utilized this book with your students, please try to find a copy!  It’s called The Doorbell Rang by Pat Hutchins.  In this story, Ma makes some cookies to be split between the kids.  Then the doorbell rings and more kids come, so the problem has to be refigured. This scenario repeats. As a class, you can duplicate the story with a different # of cookies and children.

Another great story emphasizing equal groups (as well as arrays) is the story One Hundred Hungry Ants by Elinor Pinczes.  In this story, 100 ants are on their way to raid a picnic. They start off in one straight line (1 x 100), but then rearrange into different equal groups to shorten the line (2 lines of 50, 4 lines of 25, etc.). A nice project after reading this book is to see how many ways a different given # of ants (or other animals / objects) can be divided into equal groups / rows.

 

By clicking on the links for each book above, you will be taken to Amazon for more details.

As I mentioned earlier, many children’s view of equal groups regarding division is to use manipulatives and/or draw circles / plates to match the divisor and then divide up the “cookies” equally in these groups.  Let’s say you had this problem: “There are 12 cookies to be divided onto 3 plates equally.  How many cookies would go on each plate?” As you observe the students:

  • How are they dividing up the cookies? One at a time, two at a time, randomly, trial and error?
  • Are the “cookies” scattered randomly on the plate / circle?  Or, are they arranged in an easy-to-see pattern so they are easily counted (by the student and yourself as you walk around the room)?
  • Are the students able to verbally tell you how they divided them?
  • Are the students making the connection to multiplication by noting that 3 x 4 = 12?
  • Can they solve similar problems using language other than plates / cookies?
    • Try shelves / books; trays / brownies; buildings / windows; flowers / petals; students / rows of desks, stars / # of points; aquariums / fish; boxes / donuts; etc.

Use of manipulatives of various types (cubes, tiles, counters) is important for children to have their hands on the objects being divided. This is how they work out their thinking. Then work toward paper/pencil drawings before going to the abstract use of numbers only.  This doesn’t have to be done in separate lessons, however. There is great value for children to see how the concrete, pictorial, and abstract representations all work together.

Also, help children list synonyms for the dividing process:  distribute, share, split, separate, halve, quarter, partition

Here are a few strategies I believe help make the equal groups process more efficient: Continue reading

Geometry Part 4: More Composing and Decomposing

by C. Elkins, OK Math and Reading Lady  

There are so many good ways to help students compose and decompose shapes (2D and 3D), so I will focus on some more by using tangrams and 2D paper shapes. In case you missed it, my last post focused on ways to use 1″ color tiles and pattern blocks to compose and decompose shapes. Click HERE to link back to that.

  1. Give students paper shapes of these polygons:  rectangle, square, hexagon, trapezoid, rhombus. Click here for a FREE pdf copy: Decompose and Compose Polygons.
    • Students should color each paper shape one solid color (a different color for each shape). My advice is to use light colors because they will be drawing lines on the shapes and light colors enable them to see the lines.
    • Model how to draw 1 or 2 lines to decompose the shape into smaller shapes.  For first and 2nd grade, I recommend you show them how to use at least one corner of the shape to connect to another corner or side using a straight edge or ruler. This way the newly created shapes will resemble ones they already know (triangle, trapezoid, rectangle). Older students can be given a little more leeway — their decomposing may result in other more irregular polygons. Here is one way to decompose.
    • Cut apart on the lines. Have students put their initials or name on the back of each piece (in case it gets separated or ends up on the floor).
    • Each student puts their cut-up pieces in a baggy for safe-keeping. Then the student can take them out and try to compose them back into their original shapes.  This is where the color-coding comes in handy (all the yellow go together, all the green, etc.).
    • Students can trade their baggies with others to compose their shapes.
    • When students are done with the shape puzzles, they can glue them back together on background construction paper (or take them home for practice, or keep at school for ongoing work).
    • Discuss together how many different ways these shapes were decomposed using 1 or 2 lines.
  2. Use the book, “The Greedy Triangle” by Marilyn Burns as a springboard to compose other polygons using various numbers of triangles.  In this book, the triangle keeps adding a shape to himself (after a visit to the “Shapeshifter”). There are many good pictures in this book illustrating common things with the named shape.  This is also a great way to connect art to math. You can start with squares which the students must cut in half on the diagonal, or start with pre-cut triangles. Length of edges must match. Level 0 students can just try out different combinations. Level 1-2 students would analyze the properties more and name the new shapes. You can even emphasize symmetry (as I have shown with the bottom row). Here is the link to the full article about this wonderful activity. Math Art: The Greedy Triangle Activity

Continue reading

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

First Day Math & Literature Activity K- 5

by C. Elkins, OK Math and Reading Lady

The book, Chrysanthemum, by Kevin Henkes is one my my all time favorite first-day-of-school stories to share with my students – no matter what grade level. The main character is Chrysanthemum, who is all excited about her first day of school until the other students start making fun of her name because it is soooo long. This makes her reluctant to go to school until everyone finds out their favorite music teacher has a long name (Delphinium) and is planning to name her new baby Chrysanthemum. A poignant story to help children develop a sense of empathy and compassion and realize that everyone’s name is special – no matter what it is or how long or short it is!

Math Connection Grades K-2

  • Letter and name recognition
  • Counting letters in names
  • Name graph with a variety of methods (paper graph, color tile or unifix cube graph, etc.)
  • Name grid art activity (see below)
  • Comparing name lengths

Math Connection Grades 3-5

  • Name graph – can use first, middle, and/or last names. To start, just have students write their name on a post-it-note and stick it on the board. Then rearrange into columns or rows according to how you are collecting your data. Or make a frequency table, line plot, percentage pie chart, etc.
  • Name grid art activity (see below). Review terms: row, column, grid, array.
  • Use some type of strategy to determine total number of letters in first names in the class (repeated addition, multiplication). Using the example graph, students could add 3 + (4 x 5) + (5 x 8), and so on. Let students think of the strategy though!
  • Determine most often and least often used letters.
  • Determine the mean, median, mode, and range using length of names.

Name grid art activity Continue reading