• The Solving Equations worksheet (M-3-5-2_Solving Equations and KEY.docx) may be used to determine how well students can solve for a variable in multiplication and division equations.
• The Wipe Out! Activity (M-3-5-2_Wipe Out! and KEY.docx) may be used to determine how well students know and understand four-fact families.
• Also, use the Lesson 2 Exit Ticket (M-3-5-2_Lesson 2 Exit Ticket and KEY.docx) to quickly assess student understanding of solving multiplication and division equations.

Four-Fact Families

Introduce families of multiplication and division facts. There are always four facts in each family unless the factors are the same such as ; in this case there are only two facts in the family. Use the following examples to introduce students to four-fact families.

“Taya knows that . Jeremiah says if Taya knows this fact, she can also know three other facts that are in the same fact family. Do facts really have families? What is a fact family?”

Students are likely to recognize the related multiplication fact . Be sure to write all of the related facts on the board. If students are not able to state the related division facts, write
24 ÷ ___ = ___ on the board. Ask students if they can use this clue to write the related division facts. Prompt them as needed. Be sure the four-fact family is written on the board when finished.

Introduce the following “fact family” visual:

“Using a triangle like this one to organize your fact families can be a good visual.”

Work through at least one more example with students. “Taya also knows . What other facts can Taya know that are in this fact’s family?” Students will likely be more able to state the other facts, as this is the second example. Yet, write the first number and the operation on the board if necessary to prompt students. For example, write 7 × ___ = ___ if students are struggling to identify a specific fact in the family. Be sure to show the triangle and write each fact in the four-fact family on the board.

Now, help students examine the four-fact families and look for patterns. “How many facts are in each family?” (4) “Since there are 4 facts in each family, these are called four-fact families. Look at the four-fact families from both examples. What do you notice about these four-fact families?” Students may state many observations. The goal is to be sure they notice there are two multiplication facts and two division facts in each family.

Distribute and introduce the Four-Fact Families worksheet (M-3-5-2_Four-Fact Families and KEY.docx). “Notice on the front of the practice worksheet, the four-fact families are listed but many numbers are missing. Be sure to fill in the blanks in each four-fact family. On the back of the worksheet, one fact is provided. Please write the other three facts in each four-fact family.”

As students are working, monitor their progress. Help students to understand there are two multiplication facts and two division facts in each family. Refer to the discussion (using color tiles) of the commutative property of multiplication. This should help students understand why there are two multiplication facts in each family. Also, use the discussion of multiplication and division from Lesson 1 to help students remember that multiplication finds the total number of objects in many equal groups, whereas division is used to divide the total number into many equal groups. For this reason, the product or answer of the multiplication problem is the dividend or first number in the division problems.

When students are finished working, ask individual students to write the fact families from the back page on the board. Again, help students verify that there are two multiplication and two division problems in each family, and that all four facts use the same three numbers.

Solving Multiplication and Division Equations

Distribute and display the Wipe Out! worksheet (M-3-5-2_Wipe Out! and KEY.docx) for students to observe. “What do you notice about these problems?” (There is a number missing in each number sentence or equation.)

Ask students, “With the person sitting next to you, try to determine what number is missing in the first number sentence, . Discuss what strategies you used to find the missing number.” Give students time to discuss this. Ask for volunteers to share their ideas. If students do not discuss division as a possible solution strategy, explain that division () can be used to find the missing number.

Help students understand how to record their answer as the value of , writing  = 3 in the second column. Also, help students understand how to check their answer. In the third column, write the number sentence or equation substituting the value of , resulting in 3 × 8 and compute to be sure that 3 × 8 = 24.

Now ask students, “With the person sitting next to you, try to determine what number is missing in the second number sentence, . Discuss what strategies you used to find the missing number.” Again, give students time to discuss this. Ask for volunteers to share their ideas. If students do not discuss multiplication as a possible solution strategy, explain that multiplication (5 × 6) can be used to find the missing number. Write both 5 × 6 = 30 and 30 ÷ 6 = 5 on the board. Help students understand the connection between this activity and the activity focused on four-fact families. Explain to students that if a number is missing in a division problem, as in this example, a related multiplication fact can be used to identify the missing number.

Be sure students record their answer, writing  = 30 in the second column. Also, help students check their answer, writing 30 ÷ 6 = 5 in the third column, and then verify that this is true.

Ask students to continue working in pairs to complete the Wipe Out! Practice Worksheet.

Introducing Variables

“In the Wipe Out! Problems, a number was missing in the number sentence or equation. It looked like someone scribbled out a number using a marker.” Write one example on the board: . “Instead of an empty box, mathematicians use variables to represent missing numbers in number sentences or equations. A variable is a letter or symbol that is used to represent a missing number. Any letter or symbol can be used as a variable.” Ask a student to tell you his/her favorite letter of the alphabet. Rewrite the equation using that letter as the variable, for example:  (Try to transition from the language number sentence to equation. Explain that since these statements include variables and not strictly numbers, they are more often called equations. Help students by underlining “equa” in equation. Equations are statements with equal signs, and both sides of an equation are equal in value.)

Ask at least two more students to name their favorite letters of the alphabet. Rewrite the equation using those letters. Ask students to determine the value of the missing number. Help students understand that the value of the missing number is 48, regardless of whether  or a letter is used as the variable. Help students check to be sure that 48 does make the equation true.

Provide two more examples. Write  and   on the board. Ask students to rewrite both equations using variables and find the values of the variables. Be sure to remind students to check their answers. After students are finished working, ask two volunteers to write the equations and the solutions on the board.

Distribute copies of the Solving Equations worksheet (M-3-5-2_Solving Equations and KEY.docx). Students should complete this individually. Monitor student progress.

If students are struggling, help them to think of the related operations of multiplication and division and rely on the four-fact families. For example to solve , help students instead think of , as students are more able to recall multiplication facts than division facts.

Extension:

Use the suggestions in this section to tailor the lesson to meet the needs of students. The Routine section provides suggestions for reviewing lesson concepts throughout the school year. The Small Group section is intended for students who may benefit from additional practice. The Expansion section includes a challenge opportunity for students prepared to move beyond the requirements of the standard.

• Routine: To help students review this concept, use the Match Game (M-3-5-2_Match Game.docx). The Match Game includes equations and solutions printed on individual cards. There are two ways the game can be played.

1. Students should shuffle the cards and place them face down on a table. Students take turns flipping over two of the cards. If the cards are a match, an equation and its solution, the student keeps those cards and takes another turn. If the cards are not a match, the next student takes a turn.
2. Students each draw 5 cards. Students take turns asking one of the other players for a matching card, such as “Player 2, do you have an equation that has a solution of n = 8?” Or they can ask, “Do you have the answer to 4 × n = 32?” If Player 2 does have that card, s/he gives it to the player who asked. If Player 2 does not have that card, the other player picks a card from the pile, and then the next player takes a turn.

• Small Group: Students who need additional practice may be pulled into small groups to work on four-fact families and use these to solve multiplication and division equations.

Print the fact family cards from this site: http://www.mathcats.com/explore/factfamilies/multinfo.html

Hide one number on each fact family card. Ask students to identify the missing number. Uncover the number to help them determine if they are correct. Occasionally stop at a particular card and ask them to write all four facts in the fact family. When students seem quite successful in identifying the missing number, write an equation for a particular card. Ask them to solve the equation. Help them to understand that identifying the missing number is indeed solving the equation. For example, if 3 and 15 are shown and 5 is hidden, write the equation 15 ÷ n = 3  or n × 3 = 15. Write some multiplication and some division equations when using the cards so students can practice solving both. Eventually you can ask students to write the equations and solve them.

• Expansion: Students who are ready for a greater challenge can write their own real-world problems. Students should work in pairs. Each student creates a multiplication or division equation using a variable. Students in each pair then exchange the equations. Each student creates a real-world problem for the equation and solves the word problem. Students then exchange their word problems for their partner to check the accuracy and provide immediate feedback.