<p>The main objective of this lesson is to use inverse operations to solve multiplication and division equations. The lesson commences with an examination of families comprising four multiplication and division facts to&nbsp;bolster this goal. Students are going to:&nbsp;<br>- determine every division and multiplication fact in a family of four facts.&nbsp;<br>- solve for an unknown quantity in a division or multiplication equation that is represented by a variable.</p>

<p>- What are the mathematical representations for relationships?&nbsp;<br>- What are some applications for expressions, equations, and inequalities in the quantification, modeling, solving, and/or analysis of mathematical situations?&nbsp;<br>- How does effective communication benefit from mathematics?&nbsp;<br>- How do we represent, compare, quantify, and model numbers using mathematics?</p>

<p>- Division: The mathematical operation of splitting a quantity into equal groups. (For example, 8 ÷ 2 = 4 because splitting 8 into 2 equal groups results in 2 groups of 4.)&nbsp;<br>- Equation: A statement of equality between two mathematical expressions.&nbsp;<br>- Factor: A number that is multiplied with another number to form a product.</p>

<p>- One copy of the Four-Fact Families worksheet (M-3-5-2_Four-Fact Families and KEY) per student&nbsp;<br>- One copy of the Wipe Out! worksheet (M-3-5-2_Wipe Out! and KEY) per student&nbsp;<br>- One copy of the Solving Equations worksheet (M-3-5-2_Solving Equations and KEY) per student&nbsp;<br>- Copies of the Match Game (M-3-5-2_Match Game) as needed. Cut cards apart before playing.&nbsp;<br>- One copy of the Lesson 2 Exit Ticket (M-3-5-2_Lesson 2 Exit Ticket and KEY) per student</p>

<p>- The Solving Equations worksheet (M-3-5-2_Solving Equations and KEY) can be utilized to assess students' proficiency in solving multiplication and division equations involving variables.&nbsp;<br>- You can assess students' knowledge and comprehension of four-fact families by using the Wipe Out! Activity (M-3-5-2_Wipe Out! and KEY).&nbsp;<br>- Additionally, you can rapidly evaluate students' comprehension of solving multiplication and division equations by using the Lesson 2 Exit Ticket (M-3-5-2_Lesson 2 Exit Ticket and KEY).</p>

<p>Formative assessment, modeling, scaffolding, and active engagement&nbsp;<br>W: Students will be able to construct four-fact families and recognize related multiplication and division facts. After that, students will learn about variables and solve equations involving multiplication and division. There will also be practice at determining whether the answers are reasonable.&nbsp;<br>H: Grab students' attention by emphasizing that mastering one fact—multiplication or division, for example—actually enables them to learn all four facts. Next, use the Wipe Out! problems with one missing number to keep students engaged. Ask students to come up with ideas on how to locate the absent number.&nbsp;<br>E: Give the students one fact. Ask them to locate the other facts in the fact family. Students should finish the Four-Fact Families worksheet in pairs. Encourage students to discuss methods for figuring out a missing number or variable, and provide an example of how to double-check or confirm their answers.&nbsp;<br>R: To review equation solving techniques, students will either complete the Solving Equations practice worksheet in class or at home. Students can practice solving equations by using the matching-equations cards.&nbsp;<br>E: Use the practice worksheet for solving equations to gauge students' comprehension, and keep track of their progress as they play the match game.&nbsp;<br>T: You can adapt the lesson to fit the needs of your students by using the ideas in the Extension section. Students who are prepared for a challenge beyond the requirements of the standard as well as those who might benefit from more practice are both given suggestions.&nbsp;<br>O: The main goal of this lesson is to get students to solve equations involving multiplication and division for missing numbers or variables. To&nbsp;help students grasp multiplication and division as inverse operations, the lesson is scaffolded, starting with four-fact families. After that, students apply their knowledge of related facts to solve division and multiplication problems.</p>

<p><strong>Four-Fact Families</strong><br><br>Provide families of division and multiplication facts. Every family consists of four facts, unless all the factors are the same, such as 7 × 7 = 49; in this case the family consists of only two facts. Introduce four-fact families to your students using the following examples:&nbsp;<br><br><strong>"Taya understands that 3 x 8 = 24. 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? Describe a factual family."</strong><br><br>The related multiplication fact 8 × 3 = 24 is probably familiar to students. Make sure you write down all related facts on the board. If students are not able to state the related division facts, write<br>24 ÷&nbsp;___ = ___ on the board. See if the students can write the relevant division facts using this hint. As necessary, prompt them. When you're done, make sure the four-fact family is written on the board.<br><br>8 × 3 = 24<br>3 × 8 = 24<br>24 ÷ 3 = 8<br>24&nbsp;÷ 8 = 3<br><br>Introduce the following “fact family” visual:<br><img src="https://storage.googleapis.com/worksheetzone/images/Screen Shot 2024-04-02 at 17.14.16.png" width="297" height="138"><br><strong>"Organizing your fact families with a triangle like this one can be a helpful visual."</strong><br><br>Go over at least one more example with the class.<strong> "Taya understands that 63 ÷ 7 = 9. What other facts that belong to the same family as this one can Taya know?"</strong> Since this is the second example, students will probably be able to state the other facts more easily. However, if needed, prompt the class by writing the first number and the operation on the board. For instance, if students are having trouble identifying a particular fact in the family, write 7× ___ = ___. Make sure to display the triangle and write on the board each of the four facts in the family.<br><img src="https://storage.googleapis.com/worksheetzone/images/Screen Shot 2024-04-02 at 17.14.40.png" width="283" height="137"><br>Now help students in analyzing and identifying patterns in the four fact families. <strong>"In each family, how many facts exist?”</strong> (4) <strong>"These are known as four-fact families because each family consists of four facts. Examine the four-fact families found in the two cases. How do these four-fact families catch your attention?"</strong> Students are free to make many observations. The goal is to make sure they recognize that every family has two division facts and two multiplication facts.<br><br>Give out the worksheet on Four-Fact Families (M-3-5-2_Four-Fact Families and KEY) and explain it to the students. <strong>"Observe that although the four fact families are listed on the front of the practice worksheet, many of the numbers are missing. Make sure to complete every four-fact family. There is one fact listed on the worksheet's reverse. In each four-fact family, please write the remaining three facts."</strong><br><br>Watch the progress of the students while they are working. Teach students that every family contains two division facts and two multiplication facts. See the explanation of the commutative property of multiplication (with color tiles). Students should be better able to comprehend why each family contains two multiplication facts as a result of this. Additionally, utilize the lesson's discussion on division and multiplication to remind students that division is used to divide the total number into many equal groups, whereas multiplication finds the total number of objects in many equal groups. As a result, the dividend or first number in division problems corresponds to the product or answer of the multiplication problem.<br><br>As soon as each student has completed their work, have them write the fact families from the back page on the board. Remind students that each family contains two division and two multiplication problems and that the three numbers used in all four facts are the same.<br><br><strong>Solving Multiplication and Division Equations</strong><br><br>Give the students the Wipe Out! worksheet (M-3-5-2_Wipe Out! and KEY) to look over. <strong>"What have you observed about these issues?"</strong> (Every number sentence or equation has one missing digit.)<br><br>Instruct students to identify the missing number in the first number sentence by working with the person seated next to them, ... x 8 = 24.<strong> "What methods did you employ to locate the missing number? Let students talk about this for a while." </strong>Invite volunteers to discuss their ideas. You can use division (24&nbsp; ÷ 8) to find the missing number if students do not bring up division as a potential solution strategy.<br><br>Assist students in understanding that putting... = 3 in the second column represents the value of their response. Assist students in understanding how to double-check their answers. Write the number sentence or equation in the third column, changing the value to 3×8, and then compute to confirm that 3× 8 = 24.<br><br>Present the question to the class now: <strong>"Identify the missing number in the second number sentence, …&nbsp;÷ 6 = 5, by working with the person seated next to you. What methods did you employ to find the missing number?" </strong>Give the students time to talk about this once more. Invite volunteers to contribute their thoughts. Remind students that multiplication (5 × 6) can be used to find the missing number if they do not bring up multiplication as a potential solution strategy. On the board, write 5 × 6 = 30 and 30 ÷ 6 = 5. Assist students in realizing how this exercise relates to the four-fact family activity. Inform students that a related multiplication fact can be used to determine the missing number in a division problem, like in this example.&nbsp;<br><br>Make sure that the students note their response in the second column by writing … = 30. Assist students in double-checking their answer by having them write 30 ÷ 6 = 5 in the third column and then confirm that this is accurate.<br><br>Encourage students to keep completing the Wipe Out in pairs! Practice Page.<br><br><strong>Introducing Variables</strong><br><br><strong>"In the Wipe Out! Problems, the equation or number sentence contained a missing number. It appeared as though someone had written a number with a marker." </strong>Put the following example on the board: ... ÷ 6 = 8. <strong>"Mathematicians utilize variables to indicate missing numbers in number sentences or equations, rather than an empty box. A variable is a character or symbol that stands for an absent digit. A variable can be any letter or symbol."</strong> Get a student to share with you their alphabetic favorite letter. Rewrite the equation as follows, using that letter as the variable: s ÷ 6 = 8. (Aim to move from the numerical sentence in language to the equation. Justify the term "equations"; these statements are more commonly used because they involve variables rather than just numbers. Underline "equal" in the&nbsp;equation to assist students. Equations are expressions with equal signs, and the values on both sides of an equation are the same.)<br><br>Get at least two more students to list their alphabetic favorites. Use those letters to rewrite the equation. Students should be asked to calculate the value of the absent number. Aid students in realizing that, regardless of whether a letter is used as the variable or .... , the missing number's value is 48. Assist students in verifying that the equation is true when 48 is present.<br><br>Give another two instances. On the board, write 63÷ ... = 7 and ....&nbsp;÷ 3 = 2. Students should find the values of the variables and rewrite both equations using variables. Make sure to remind students to double-check their responses. Once the students have completed their work, have two volunteers write the solutions and equations on the board.<br><br>Make copies of the M-3-5-2_Solving Equations and KEY&nbsp;worksheets on solving equations. Each student should finish this on their own. Track the development of your students.<br><br>If students are having trouble, assist them by using the four-fact families and the related operations of division and multiplication. Because students can recall multiplication facts more easily than division facts, help them think of 7 x j = 63 when solving problems like 63 ÷ j = 7.<br><br><strong>Extension:</strong><br><br>Make sure the lesson is student-centered by modifying it according to the advice in this section. Throughout the academic year, the Routine section offers ideas for reviewing the lessons' concepts. Students who could use more practice are targeted for the Small Group section. For students who are ready to go above and beyond the requirements of the standard, there is a challenge opportunity in the Expansion section.<br><br><strong>Routine:</strong> Use the M-3-5-2_Match Game to assist students in reviewing this concept. Each card in the Match Game has an equation and its solution printed on it. There are two ways to play this game.<br><br>1. After shuffling the cards, students should lay them face down on a table. Students turn over two of the cards in shifts. Should the cards contain a match, an equation,&nbsp;and its answer, the pupil retains those cards and advances to the next round. The following student takes a turn if the cards do not match.<br>2. Every student draws five cards. Using a phrase like "Player 2, do you have an equation that has a solution of n = 8?" Each student takes a turn asking a fellow player for a matching card. Alternatively, they could inquire, "Are you able to solve 4 × n = 32?" Player 2 hands the card to the player who asked for it if they actually possess it. If Player 2 does not have that card, the other player selects a card from the pile, and so on.<br><br><strong>Small Group:</strong> Pupils who require more practice might be divided into groups of four and work on four-fact families, using these to solve division and multiplication problems.<br><br>Print the fact family cards from this site: <a href="http://www.mathcats.com/explore/factfamilies/multinfo.html">http://www.mathcats.com/explore/factfamilies/multinfo.html</a>&nbsp;<br><br>On every fact family card, conceal one number. Ask pupils to point out the omitted number. To assist them in determining whether they are correct, uncover the number. Ask them to write all four facts in the fact family when you pause at a specific card. When pupils appear to be doing a fairly good job of figuring out the missing number, create an equation for that specific card. Have them figure out the equation. Assist them in realizing that the equation can be solved by finding the missing number. For instance, in the event that only 3 and 15 are visible while 5 is hidden, enter 15 ÷ n = 3 or n × 3 = 15. Using the cards, write some division and some multiplication equations so that students can practice solving both types of problems. You can eventually ask students to write down the equations and find solutions.<br><br><strong>Expansion: </strong>Students who are ready for a more advanced challenge can create their own real-world problems. It is recommended that students work in pairs. Using a variable, each student constructs an equation for multiplication or division. After that, students in each pair switch out the equations. For the equation, each student formulates a problem from the real world and works through the word problem. After that, students trade word puzzles with a partner so that they can verify their accuracy and give prompt feedback.</p>