# Multiplication strategies — Equal groups

Thanks for checking in on another multiplication strategy! The focus for this post 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. Be sure to see the links at the end for books and my free equal groups story cards.

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
• 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:  Here are some story problem task cards and templates for solving multiplication and division problems using the equal groups strategy. Click to see the blog post on equal groups story problems and get my FREE set of story problem cards:  HERE

Enjoy!!  Many of you are now off for a well-deserved summer break. Use your summer time to catch up on my postings from this past year, or email me for more information. Also, feel free to comment on any article about your experience or additional tips.

# Math Art Part 2: Decomposing and composing squares and triangles

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

# First Day Math & Literature Activity K- 5

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!

• 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