A Geometric Model for Multiplication
Grade 4
-adapted from “Candy Boxes” activity by Marilyn Burns
About Teaching Mathematics  page 197-198

Objective:
       Today we will make rectangular constructions in order to analyze patterns to make recommendations about candy box sizes.

Materials:
	1 set of mixed polygons that includes several different rectangles and a square 
            ruler, scissors, glue stick, centimeter graph paper available for each pair/team
            baggie with 24 centimeter cubes or tiles
           post-it centimeter chart paper-1to 2 sheets per pair or grid paper to match the tiles
           markers   
           investigation scenario, research problem (directions)

Before

1.	Give students a set of polygons and have them locate the rectangles.  Ask them “How did you decide?” and record it on post-it.  Whole group share and teacher record.  You would hopefully get  some of this information: characteristics of rectangles, a square is a rectangle, how to find/label the dimensions, area
2.	Arrange the students into pairs or teams.  Distribute the centimeter cubes.
3.	Display the investigation scenario on the document camera or read it orally.  
         
 “Candy Boxes”
Pretend that each centimeter cube is a piece of candy that is sold in rectangular boxes that are always packed in just 1 layer.  Sampler boxes have 6 pieces in each box.  Take 6 centimeter cubes and arrange them in as many ways possible to show what the sampler boxes could look like.  Then record each on your grid paper and cut them out.  Think about how you can be sure that you have all of the possible boxes.

    Share their results using the cut outs and the document camera.  Teacher records on chart paper for wall display being sure to label the length and width.  Be sure to make a connection to the area of each rectangle.  Record the dimensions as expressions on the chart.  Example: 1 x 6; 3 x 2; 2 x 3; 6 x 1

During

1.	Pose the research problem:  

“Candy Boxes”
Each pair/team of you have been hired by a candy company as a design research team.  The president of the company has asked for a report about the different boxes possible for different amounts of candies to use for holiday boxes.  For each number of candies your pair/team has been assigned, you need to:
1.	Use the centimeter cubes to find all of the possible rectangles.
2.	Draw and cut out a representation for each box.
3.	Record your results on the chart paper to use in your presentation to the president in a similar format to the one the class did for 6 candies.
4.	Write a summary of your findings and a recommendation for the box you think best.

   2.  Assign each group 1 or 2 amount of candies up to 24.  

After
1.	Place the charts around the room and get the students to discuss the patterns they see in their pairs/teams.  Record them on the original 6 candy grid paper.  Share results and teacher records for the class.  Hopefully they will see or you can lead them to see: the number of candies (total) is a product, the dimensions are factors, 1 is a factor or all of them, commutative property, maybe prime/composite with or without the vocabulary, square numbers, numbers with common factors
2.	With the class complete charts for the missing products 1-24 so you have a complete set for a wall display that can be a reference for many skills. 
3.	Have the students use their cut-outs in their own notes.  I plan to have them use a page in their math log (composition book) for each poster so they have a personal reference to use when not in my classroom.

Assessment Ideas
1.	Give them a number larger than 24 like 36 and have them list everything they know about the number.
2.	Have them repeat the activity for a larger number and write about the patterns.
3.	Place the numbers in a Triple Venn  using the headings that were brought out in the discussion.  For example: Odd or even, factors of _____, Composite/prime

Differentiation/Follow-up Ideas
1.	Identifying and describing patterns in 1-100 or 0-99 charts.
2.	Continue with numbers larger than 24 or multiples of certain numbers to identify patterns.
3.	Use the arrays or candy boxes to develop/review the concepts of perimeter and area.  This addition might make richer justifications for the box they recommend to the president of the company.
4.	Add layers to the boxes and use centimeter cubes to develop the concept of volume.