Solve Open Sentences with Multiplication and Division Student Probe What number can replace in each of the following s to make a true statement? 5 x 7 = 18 3 = x 9 = 4 x 9 36 4 = 4 6 = 48 6 Lesson Description This lesson helps develop student understanding that symbols and letters can represent a specific number or numbers in s. The teacher may ask the student, What goes in the box to make the sentence true? As the symbol is replaced with a letter, the teacher asks, What number could stand for the letter to make the sentence true? Rationale Students are expected to write s (equations) and find solutions. Initially, students have difficulty understanding that the symbols represent a missing or unknown value. Initially, work with finding the variable that makes the sentence true solving the equation should rely on relational thinking. The use of variables, whether symbols or letters, is a powerful representational device that allows for the expression of generalizations. The ultimate goal is for students to work with expressions involving variables without thinking about the specific number or numbers that the letters may stand for. Preparation At a Glance What: Determine the unknown whole number in a multiplication or division equation Common Core State Standard: 3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Mathematical Practices: Make sense of problems and persevere in solving them. Look for and express regularity in repeated reasoning. Who: Students who cannot determine the missing value, represented by a symbol or letter, in a. Grade Level: 3 Prerequisite Vocabulary: none Prerequisite Skills: meaning of multiplication and division, understanding of inverse relationship of multiplication and division Delivery Format: small group, individual Lesson Length: 30 minutes Materials, Resources, Technology: none Student Worksheets: none
Lesson The Expect students to say or do If students do not, then the 1. What number can we put 24 Use smaller numbers; such as 2 x 3 = 6 x 4 = true? Rewrite the How do you know? Because 6 x 4 = 24 6 x 4 = 2. What number can we put 5 x 3 = x 3 true? How do you know? 5 Because 5 x 3 = 5 x 3. 6 x 4 = 24 Use a balance scale. Place cubes in the left pan, counting as 1 group of 3, 2 groups of 3, 3 groups of 3, etc. to 5 groups of 3. Read the equation as 5 groups of 3 is the same as how many groups of 3? Let s see how many groups of 3 cubes we need to add to the right pan for the balance to be level. Continue to add groups of 3 one at a time, with students counting as each group is put in the pan, until the balance is level. How many groups of 3 did we put it? (5) If we replace the with 5, then is this a true statement? 5 groups of 3 is the same as 5 groups of 3 (yes) Rewrite the 5 x 3 = x 3 5 x 3 = 5 x 3
The Expect students to say or do If students do not, then the 3. What number can we put x 8 = 4 x 8 true? How do you know? 4 Because 4 x 8 = 4 x 8 Use a balance scale. Place cubes in the right pan, counting as 1 group of 8, 2 groups of 8, 3 groups of 8, 4 groups of 8. Read the equation as follows: How many groups of 8 is the same as 4 groups of 8? Let s see how many groups of 8 cubes we need to add to the left pan for the balance to be level. Continue to add groups of 8 one at a time, with students counting as each group is put in the pan, until the balance is level. How many groups of 8 did we put it? (4) If we replace the with 4, then is this a true statement? 4 groups of 8 is the same as 4 groups of 8 (yes) Rewrite the x 8 = 4 x 8 4 x 8 = 4 x 8
The Expect students to say or do If students do not, then the 4. Repeat with division s. What number can we put 28 4 = true? 7 Teacher uses smaller numbers; such as 12 6 = Rewrite the numbers sentence below the open : 5. What number can we put 48 6 = 48 true? 6. What number can we put = 36 9 true? 7. Do additional multiplication and division problems as needed following steps above. 28 4 = 28 4 = 7 6 Teacher will take 48 cubes and divide them into 6 equal groups. Give the student 48 cubes. Have him divide them into equal groups so that it looks just like the teachers. Ask student if the two groups are equal (yes). Line them up to show they are the same. Write the equation 48 6 = 48 6 4 Change the problem around to 36 9 = Ask student What goes in the box to make the number sentence true? (4) Change the problem back like it was first, and ask student what goes in the box. (4) If student does not understand, refer to Equals Means the Same As.
Teacher Notes: Variations Formative Assessment What number can replace statement? in each of the following s to make a true 8 x 8 = x 7 = 6 x 7 18 2 = 54 9 = 9 9= 72 9 References Mathematics Preparation for Algebra. (n.d.). Retrieved February 25, 2011, from Doing What Works: http://dww.ed.gov/practice/?t_id=20&p_id=48 Van de Walle, John A., Karp, Karen S., and Bay- Williams, Jennifer M., (2010), Elementary and Middle School Mathematics: Teaching Developmentally, Boston, Allyn & Bacon