## Chocolate FACTORy: Finding Factors of Numbers 1 Through 36

• Lesson
3-5
2

In this lesson students create rectangular arrays to represent sizes of chocolate boxes. They find all of the factors of each number up to 36 and learn the difference between prime and composite numbers. Then they play an online game to practice finding factors for each product up to 36.

To prepare for this lesson, take 36 blank pages of 8.5 x 11 paper and number them 1 to 36. The number should be large enough to be seen on the opposite of the room, but without taking the majority of space on the page. Hang them up around the room to be used during the lesson. Also, every student will need a copy of the Chocolate FACTORy Activity Sheet.

To introduce the lesson, tell the students that the Gear-Are-Deli Chocolate Factory produces chocolate squares, and until now they have always sold the chocolate squares in individual wrappers. To be more cost effective, the company is looking into designing boxes that will hold different amounts of chocolates. The company only has three rules: 1) Chocolates cannot be stacked on top of each other or they might melt. 2) All boxes must be rectangular. 3)The largest box they will consider would hold 36 chocolates.

Divide the class into groups with 2-4 students in each group. Each group needs a supply of graph paper, scissors, and a roll of tape. Tell the students that the company would like to see how the boxes would look, so they need to make as many different models for each number of chocolates as possible. The students are to trace each box onto graph paper, cut them out, and tape them to the correct numbered page around the room. For example, 6 chocolates could have a box that is 1 by 6 or a box that is 2 by 3. Each square on the graph paper represents 1 chocolate. A box that is 3 by 2 is the same as a box that measures 2 by 3, so only one of these boxes should be included. The 2 factors used to create the box should be labeled on the number page next to the sides of the box so students can easily see them for each box.

Note: If this activity is to be completed in 45 minutes or less, groups of students should be assigned numbers so there is no duplication.

Once the groups have finished creating their boxes they should circulate the room and look at the numbered pages to see if any possible boxes have not been included yet. If a student sports one, then they should announce it and make that box and attach it to the page. If students are unable to find all the boxes for a particular number of chocolates, you may bring it to their attention and ask them to focus on finding the missing box.

After all the possible boxes for each number of chocolate have been made, have the students complete the Chocolate FACTORy activity sheet independently. Depending on how long it takes students to complete this sheet, it may be done as a homework assignment.

Ask students to share their observations from the Chocolate FACTORy Activity Sheet. Which numbers only had one possible box? Why did this happen? Explain that the dimensions of the boxes are called the factors of that number. For example, a 3 by 4 box holds 12 chocolates and has factors of 3 and 4. Some numbers are called prime numbers because they have two distinct factors: 1 and themselves. Tell students that by this definition, 1 is not considered a prime number although it has a factor of 1 and itself.Have students look at the numbered pages around the room. Which numbers are prime numbers? [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.]

Tell students that any number with factors other than 1 and itself is called a composite number. What are some examples of composite numbers? [16 and 24.] Which number(s) had the most number of possible boxes? [36] Which numbers have 3 as a factor? [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, and 36.] How do they know? [Students should be able to demonstrate that the numbers they chose all have boxes with 3 squares on one side.]

Demonstrate how to play the “Factor Dazzle” game on the Calculation Nation web site. This will help students reinforce the concept of factors, composite numbers, and prime factors. Remind students that they can look at the numbered pages around the room to remind them of the factors for each number, if needed. If your students are registered on the site and sign in, they can challenge each other on the game.

### Ideas for Differentiation

• Students who might be challenged by this activity could complete boxes for smaller composite numbers with 2 possible boxes, such as 2, 4, 6, and 9.
• More advanced students can be challenged to list the possible boxes for higher numbers without making the boxes. What number would have the most possible boxes using only single digit factors? How many numbers under 100 would have exactly 4 possible boxes?

Assessment Options

1. Were the students able to provide all of the possible boxes for any given number?
2. Were students able to explain their thinking on the Chocolate FACTORy activity sheet?
3. Were the students able to play the game successfully? How many wins did each student get? If a student had repeated losses, why did this happen?

Extensions

1. Have students create a graph with the number of chocolates (area) of the square boxes on the x-axis and the factor that makes the square on the y-axis. What patterns do they see? How would they expect the graph to look if they continued it to 1000 chocolates?
2. Give an explanation of the square root symbol and its meaning. Ask students to write and solve problems using square numbers and the square root symbol.
3. Have students create a larger game board, such as a 9×9 and test their strategies to maximize their score. Keep a list of good and bad first moves.
4. Review how to find perimeter with students. Then, ask students to think about the perimeter of each chocolate box that was made. For any given number of chocolates, which box has the least perimeter? Why?

Questions for Students

1. What number would have the most possible boxes using only single digit factors?

[8 and 6.]

2. Why can’t this number be larger?

[Because any 2 digit number would have a box that is 1 by itself, giving factors with double digits.]

3. What is the next higher number after 36 with only 1 possible box? How do you know?

[37, because it is prime.]

4. How many different square boxes could be made that hold 100 or fewer chocolates?

[10, this would include all of the square numbers from 1 through 10.]

5. How many different square boxes could be made that hold 200 or fewer chocolates?

[14.]

6. How about 500 or fewer chocolates?

[22.]

7. What would be the dimensions of the largest possible box holding 500 or fewer chocolates?

[Answers will vary. Sample answers: 2×250, 20×25, etc.]

8. Is there a strategy that allows you to maximize your score and win most the the games you play? If so, describe your winning strategy.

[Answers will vary.]

Teacher Reflection

• Did you have any learners with specific learning needs? What were they, and how did you adapt the lesson to accommodate them?
• Did the students fully understand how to play the Calculation Nation game, Factor Dazzle, before they began playing? If not, then how could the game instructions for Factor Dazzle be made clearer?
• Did all of the students remain actively engaged throughout the lesson? If not, what could improve their engagement?
• Were their any technical difficulties that delayed students’ ability to play the game? If so, how could these issues be eliminated next time?

### Learning Objectives

Students will:

• Cut out arrays to demonstrate factor combinations for all numbers up to 36.
• Compare prime and composite numbers.
• Play an online factor game.

### NCTM Standards and Expectations

• Describe classes of numbers according to characteristics such as the nature of their factors.
• Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

### Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

• CCSS.Math.Content.4.OA.B.4
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 2, Algebraic Thinking

• CCSS.Math.Content.2.OA.C.4
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Grade 2, Geometry

• CCSS.Math.Content.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP7
Look for and make use of structure.