## Times Square: Reinforcing Multiplication Skills Using Factors and Strategy

- Lesson

In this lesson, students use their previous knowledge of multiplication to identify factors and form products. Students will use Illuminations’ Times Table to identify various patterns in a multiplication table. They will then play the Multiple Factors Game and Times Square to reinforce their understanding of factors and multiples.

This lesson is most appropriate for students who understand
the process of multiplication and are beginning to recognize the relationship
between factors and products. Prior to the
lesson, check computers for access to Illuminations and Calculation Nation^{®}.
Make sure all students have joined Calculation Nation and know their screen
name and password. Note that students can enter Calculation Nation via the
“Guest Pass” option, but that will limit them to one-player games against the
computer. To participate in two-player games, they will need to register for an
account.

Project Illuminations’ Times Table to the whole group and encourage them to look for patterns. Some students may notice that multiplication is commutative [The product of 7×1 is equal to the product of 1×7, 7×2=2×7, etc.]. They may also be able to notice that if you uncover all the multiples of two, then a striped pattern appears. You may have to guide students to find more difficult patterns. For example, have students choose numbers along a diagonal path from 0×0 to 9×9 to see that the products are all square numbers. You could also ask students if they can see the relation between the products of 4 and 8. [If you multiply all of the products of 4 by 2, you will find all of the products of 8.] Discuss these patterns together as a class.

Have students work with partners (mixed ability) on computers to explore their own patterns. How are they similar to what the class observed? How are they different? Did they find any new patterns? Encourage students to share what they observe.

Multiple Factors Game Overhead

Students will play the Multiple Factors Game to help them recognize that some products have many factor pairs and others only have a few. Before introducing the Multiple Factors Game Overhead to students, do a pre-assessment to ensure students’ understanding of factors and products. Orally review basic multiplication facts by giving products and allowing students to identify the factors. Project a product on the overhead and have students write down one pair of factors for that product. [Good products to use: 12, 18, 24, 36.] Encourage students to compare and discuss their choice of factors with a partner. Using the overhead product, share the list of factors presented by the class. When you are satisfied that students can give examples of products with understanding, introduce the Multiple Factors Game.

The best way to explain the rules is to play a game against the class. Project the game board overhead for students to see. Here is an example:

1 | 2 | 2 | 2 | 2 | 3 | 3 |

3 | 3 | 4 | 4 | 4 | 4 | 5 |

5 | 5 | 5 | 6 | 6 | 6 | 7 |

7 | 7 | 8 | 8 | 9 | 9 | 10 |

10 | 11 | 12 | 13 | 14 | 15 | 16 |

17 | 18 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 30 | 32 | 33 |

34 | 35 | 36 | 40 | 42 | 45 | 50 |

Before starting the game, discuss the commutative property of multiplication (when 2 numbers are multiplied together, the product is the same regardless of order). Ask students if they can find any products on the game board by multiplying different factors.

[ 1×4=4 & 2×2=4; 1×6= 6 & 2×3=6; 1×8= 8 & 2×4=8; 1×9= 9 & 3×3=9; 2×6=12 & 3×4=12; 2×8=16 & 4×4=16; 2×9=18 & 3×6=18; 3×8=24 & 4×6=24; etc.]

Afterwards, explain the directions and play a few rounds using one color counters for the teacher and another color for the class, completing the overhead Game Board.

- Player 1 chooses a 2-digit number. They write their number on their sheet and cover the number on the game board. This is their total number of points for the round.
- Player 2 covers all the factors of Player 1’s number. He/she must find the sum of these factors, using scrap paper if necessary, and record the total as points for this round.
- If Player 2 misses any factors, Player 1 may cover them and add the points to their total.
- When a player chooses a number, the other player must be able to earn points, or the player choosing loses their turn.
- Player 2 then chooses a 2-digit number and Player 1 must cover all the factors of Player 2's number that are not already covered.
- The game will continue following the same procedure until all the possible factors and products have been covered. Players total their scores and the player with the highest score wins.

Example for completing the Score Card Activity Sheet:

Round | Points | Total Points |

1 | 1+36+2+18+3+12+4+9+6 | 91 |

2 | 36 | 36 |

Distribute one Multiple Factors game overhead and two different colored counters to each pair of students. Each student will need their own Score Sheet. Make sure each student in the pair has a pencil and a set of two different colored counters. Students play the game with a partner following the same directions explained for the whole group activity. When they are finished, if time allows, they may play the same partner, or switch partners and play again. Discuss the game with the class soliciting explanations and understanding of the relationship between factors and multiples.

Calculation Nation: Times Square

Using a projector, show students the Times Square game on Calculation Nation. Play together as a class to ensure students understand the strategy of the game. Discuss how this game relates to the Multiple Factors Game [Many numbers can be made with different factor pairs, while others can be made with only a few.]. In this game, all possible products are listed in a table. Below the table are factors that can be multiplied to find the products. Students will play several rounds of the game. They may challenge themselves first and then play against others.

Conduct a discussion with students about which factors they chose to use when playing the game.

Sample guided question: "How did you determine which factors to use?"

[I looked for products with the least amount of factors to start. I wanted to keep the numbers with more than one factor pair for future turns when I might be limited in the factors I can choose]

### Idea for Differentiation

Allow students to use tools when needed such as: manipulatives, calculator, and peers.

- Computer and Internet access
- Times Table Applet
- Calculation Nation: Times Square
- Multiple Factors Game Overhead
- Score Card Activity Sheet

**Assessment Options **

- While students are working, monitor on-task behavior and difficulty in fact recall. Also note what strategies are being used to complete the multiplication sentences. Collect and review the completed multiplication tables, identifying amount of table completed and compare with anecdotal notes.
- Throughout the lesson, circulate, observe and question the students as they play the Multiple Factors game. Use the questions from Questions for Students to assess students’ understanding.
- Observe students’ ability to win the Times Square. Are they able to successfully block their opponents? Do they utilize any strategies? Are they able to explain their strategies?

**Extensions **

- As an extension of this lesson, students can use the game boards to determine which products have more than two sets of factor pairs. Discuss how this can be used as a strategy in playing Times Square.
- Students can play Times Square again on the computer. Encourage them to use strategy to successfully block their opponent or win the game.
- Ask the students to play the Multiple Factors game again with a new restriction: students can only use
*prime numbers*as candidates for factors. (A number is a prime number if it has no factors other than 1 and itself). If they cover up a number that is not a factor, then the student must subtract that number from the total number of points they have. - Introduce the word
*relatively prime*. (Two numbers are relatively prime if they share no common factors except 1). Have students identify at least 3 pairs of relatively prime numbers on the game board. Have them write out all the factors of each number to show that they share no common factors except 1. - Encourage students to make their own product game boards such as the following 3x3 board:1 2 34 6 89 12 16which uses factors of 1, 2, 3, and 4; students should try to get three in a row using the same rules as stated in the lesson plan.

**Questions for
Students **

1. In finding products, is 4×5 different than 5×4?

[Although numbers appear in different order, multiplication is commutative. Thus, 4×5=5×4.]

2. Were the larger numbers always worth the most points?

[Not always. The smaller number could have more factors that give you a larger sum. An example is 45 and 36; the sum of the factors of 36 is larger than the sum of the factors of 45.]

3. How do you determine when you have found all the factors of a number?

[Answers may vary. Sample: The numbers that are not covered are not factors of the product we chose.]

**Teacher Reflection **

- Did you find it necessary to make adjustments while teaching the lesson due to students’ mastery of multiplication facts? If so, what adjustments, and were these adjustments effective?
- Did you challenge the achievers? How?

### Learning Objectives

Students will:

- Develop understanding of factors, multiples, and of the relationships between them
- Understand that some products are the result of more than one factor pair

### NCTM Standards and Expectations

- Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

- Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

### Common Core State Standards – Mathematics

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.B.5

Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.C.7

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

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.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP2

Reason abstractly and quantitatively.

- CCSS.Math.Practice.MP4

Model with mathematics.