Name of Lesson: Billy s Family has Missing Addends. Topic: Primary Algebraic Reasoning Lesson # 7 Approximate time: 2 sessions Gifted Standard and element(s): G1CG1: Convergent Thinking: Students will reason logically using induction, deduction, and abduction. d. Solve problems using logical reasoning. e. Solve algebraic equations using logical reasoning. f. Develop verbal and nonverbal communication skills to convey logical reasoning. Supports CCGPS: 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. 1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. 6. Attend to precision. Unit Essential Question(s): *How can I use the mathematical meaning of equivalence to solve problems? *How can I use variables to solve problems? How can I use patterns, rules, and relationships to solve problems? Lesson Questions: How can I use a number balance to solve problems involving equality? Lesson Summary: Students will represent and solve word problems using the number balance. Key idea: Each weight used on the number balance (which shows equality when balanced) represents a specific quantity in the problem. This representation includes both given information and the solution to the problem (originally the unknown information represented by a variable). Assessment Description/Performance Task: Constructed response Informal assessment Performance task Selected response Brief Description of Assessment: Students will solve problems using the number balance. Instructional Methods: Hook/Activator: Present this word problem to students and ask them to solve it. Fred and Jessica earned 9 marbles for their group s jar. Fred earned 6. How many did Jessica earn? Have students generate number sentences for this situation using a variable. Make sure they come up with more than one that involves addition and at least one that involves subtraction. Demonstrate how to use the number balance to solve the problem. Label each weight with the part of the problem it represents: Fred s marbles, total marbles, Jessica s marbles: j. (To label the weights, use sticky notes, correction tape, painter s tape, or write directly on the weights with a marker.) Teaching Strategy: Allow pairs of students to use the number balance to solve and/or check their answers to the problem above. Discuss various number sentences and demonstrated solutions as a
class. Give students the student sheet Under the Sea and require them use the number balance to solve or to check problems. (Whether you have students in pairs or small groups or doing this as individuals may depend on how many students you have, how many balances you have, and what other activities students need to work on.) After everyone has completed Under the Sea, discuss (either in small groups with the teacher or as a class) the solution and the various number sentences used for each problem. Allow groups to demonstrate the validity of their answers using the number balance. Under the Sea 2 is for remediation/extra practice as needed. It might also be used for a review in the future. Summary by the Learner: Tell a partner how the number balance is used as a tool. When students are ready for the assessment sheet Let Me Show What I Know, use the first question from Under the Sea to demonstrate how they are to use the pictured balance to show their thinking. Since the important point is for them to understand how each part of the problem is represented on the balance, it s important for them to be able to label the picture. 1) Billy Barracuda has found 3 shrimp. His mother wanted him to get 8 for the family s dinner. How many more does he need to find? shrimp Billy needs: w (or whatever variable was used in equation) shrimp needed for dinner shrimp Billy has 3 + w 8 Billy needs to find 5 more shrimp. Give students the assessment sheet Let Me Show What I Know to complete independently. Let Me Show What I Know 2 is for any re-assessment needed. It may also be used for review in the future. Differentiation: More capable: Assign the Extension problem on the student sheet. Have students explain their solutions and reasoning. Less capable: Have students circle or underline the part of the problem represented by the variable. Have students continue to label the weights as they solve each problem. Use blank pictures of balances (provided) for support on Under the Sea and/or Under the Sea 2. Materials for this Lesson: number balances, preferably one for each pair of students class set of Under the Sea class set of Let Me Show What I Know
(as needed) copies of Under the Sea 2 (as needed) copies of Let Me Show What I Know 2 (as needed) copies of blank pictures of balances Vocabulary for this Lesson: balance equation
Name Under the Sea Solve each problem. You may use the number balance. If you do NOT use the balance to solve it, use it to check your solution. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided 1) Billy Barracuda has found 3 shrimp. His mother wanted him to get 8 for the family s dinner. How many more does he need to find? Billy needs to find more shrimp. 2) Billy s mother bought him 10 pencils for school. There are 3 in his backpack and 4 in his desk. The rest are still in the package. How many are left in the package? There are left in the package. 3) Billy and his sister, Star, each got $8 for their birthdays. Billy spent $6 and has $2 left. Star spent $4. How much does she have left? Star has left.
4) The 3 Barracuda children each have a collection of beautiful seashells. Star has 5 shells. Billy has 8 shells. Jose has forgotten how many shells he has, but he does know that together the three children have 16 shells. How many shells does Jose have? Jose has shells. 5) Billy and his friend, Carol Crab, each have the same number of marbles, some of which are red and some of which are blue. Billy has 5 red and 7 blue. Carol has 2 red. How many blue marbles does Carol have? Carol has blue marbles. Extension: Billy s friend, Melanie Mako, ate 8 more fish than Billy ate. Together they ate 14. How many fish did each eat? Billy ate fish. Melanie ate fish.
Name Let Me SHOW What I Know Solve each problem. You may use the number balance. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided Also, draw on the picture of the balance to show how this problem could be solved using an actual number balance. Be sure to label the weights you draw to show what they represent, including writing the variable on the one representing the unknown. 1) Tina Tuna has 5 shiny pebbles. Ike Pike had 2 shiny pebbles. How many shiny pebbles did they have together? Equation: They had pebbles together. 2) Aretha Albacore grew 3 seaweed stalks in her garden. Marvin Manatee has 10 seaweed stalks in his garden. How many more does Marvin have than Aretha? Equation: Marvin has more seaweed stalks than Aretha.
Name KEY Note: Under the Sea Variables used in the key are simply convenient for reference to the problem. ANY variable may be substituted for those in the key. In all answers, the expressions on each side of the equations could be reversed. Only one version is listed in the key. For example: 3 + y 8 OR 8 3 + y Solve each problem. You may use the number balance. If you do NOT use the balance to solve it, use it to check your solution. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided 1) Billy Barracuda has found 3 shrimp. His mother wanted him to get 8 for the family s dinner. How many more does he need to find? 3 + y 8 8 y 3 y + 3 8 8 3 y Billy needs to find 5 more shrimp. 2) Billy s mother bought him 10 pencils for school. There are 3 in his backpack and 4 in his desk. The rest are still in the package. How many are left in the package? 10 3 4 g 10 g 4 3 10 g 3 4 10 g 3 + 4 10 7 g 10 4 3 g 10 4 g 3 10 3 g 4 10 g 4 + 3 10 g 7 7 + g 10 g + 7 10 10 g + 3 + 4 Addends may be in any order. If a student writes either of these: 10 3 + 4 g 10 4 + 3 g ask the child to explain his/her reasoning. If the child can explain that he/she added the two known quantities of pencils, and then subtracted the sum from the total number of pencils, he/she understands the situation. Show the child that the correct way to write this would be: 10 (3 + 4) g 10 (4 + 3) g Explain that the parentheses mean do this first. There are 3 left in the package. 3) Billy and his sister, Star, each got $8 for their birthdays. Billy spent $6 and has $2 left. Star spent $4. How much does she have left? 6 + 2 4 s 6 + 2 s 4 6 + 2 4 + s 2 + 6 4 s 2 + 6 s 4 2 + 6 4 + s 8 4 s 4 + s 8 6 + 2 s + 4 8 s 4 s + 4 8 2 + 6 s + 4 Star has $4 left.
4) The 3 Barracuda children each have a collection of beautiful seashells. Star has 5 shells. Billy has 8 shells. Jose has forgotten how many shells he has, but he does know that together the three children have 16 shells. How many shells does Jose have? 16 5 8 j 16 8 j 5 16 5 j 8 13 + j 16 8 + 5 16 - j 16 8 5 j 16 j 8 5 16 j 5 8 j + 13 16 5 + 8 16 j 16 j 13 16 13 j j + 5 + 8 16 Addends may be in any order. If a student writes either of these, 16 8 + 5 j 16 5 + 8 j ask the child to explain his/her reasoning. If the child can explain that he/she added the two known quantities of shells and then subtracted the sum from the total number of shells, he/she understands the situation. Show the child that the correct way to write this would be: 16 (8 + 5) j 16 (5 + 8) j Explain that the parentheses mean do this first. Jose has 3 shells. 5) Billy and his friend, Carol Crab, each have the same number of marbles, some of which are red and some of which are blue. Billy has 5 red and 7 blue. Carol has 2 red. How many blue marbles does Carol have? 5 + 7 2 s 5 + 7 s 2 5 + 7 2 + s 7 + 5 2 s 7 + 5 s 2 7 + 5 2 + s 12 2 s s + 2 12 5 + 7 s + 2 12 s 2 2 + s 12 7 + 5 s + 2 Carol has 10 blue marbles. Extension: Billy s friend, Melanie Mako, ate 8 more fish than Billy ate. Together they ate 14. How many fish did each eat? b + 8 + b 14 b + m 14 (Children must realized that m b + 8 for this one to work.) Students will probably solve this one by guess and check. Accept any explanation that shows understanding of the situation. Billy ate 3 fish. Melanie ate 11 fish.
Name KEY Let Me SHOW What I Know Note: Variables used in the key are simply convenient for reference to the problem. ANY variable may be substituted for those in the key. In all answers, the expressions on each side of the equations could be reversed. Only one version is listed in the key. For example: 3 + y 8 OR 8 3 + y Solve each problem. You may use the number balance. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided Also, draw on the picture of the balance to show how this problem could be solved using an actual number balance. Be sure to label the weights you draw to show what they represent, including writing the variable on the one representing the unknown. 1) Tina Tuna has 5 shiny pebbles. Ike Pike had 2 shiny pebbles. How many shiny pebbles did they have together? Tina s pebbles Ike s pebbles pebbles together: p Equation: 2 + 5 p 5 + 2 p They had 7 pebbles together. 2) Aretha Albacore grew 3 seaweed stalks in her garden. Marvin Manatee has 10 seaweed stalks in his garden. How many more does Marvin have than Aretha? Marvin s stalks Aretha s stalks how many more Marvin has than Aretha: x Equation: 3 + x 10 x + 3 10 10 x 3 10 3 x Marvin has 7 more seaweed stalks than Aretha.
Name Under the Sea 2 Solve each problem. You may use the number balance. If you do NOT use the balance to solve it, use it to check your solution. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided 1) Billy Barracuda has found 5 shrimp. His mother wanted him to get 8 for the family s dinner. How many more does he need to find? Billy needs to find more shrimp. 2) Billy s mother bought him 12 pencils for school. There are 3 in his backpack and 4 in his desk. The rest are still in the package. How many are left in the package? There are left in the package. 3) Billy and his sister, Star, each got $10 for their birthdays. Billy spent $6 and has $4 left. Star spent $7. How much does she have left? Star has left.
4) The 3 Barracuda children each have a collection of beautiful seashells. Star has 5 shells. Billy has 7 shells. Jose has forgotten how many shells he has, but he does know that together the three children have 18 shells. How many shells does Jose have? Jose has shells. 5) Billy and his friend, Carol Crab, each have the same number of marbles, some of which are red and some of which are blue. Billy has 3 red and 8 blue. Carol has 2 red. How many blue marbles does Carol have? Carol has blue marbles. Extension: Billy s friend, Melanie Mako, ate 7 more fish than Billy ate. Together they ate 17. How many fish did each eat? Billy ate fish. Melanie ate fish.
Name Under the Sea 2 Variables used in the key are simply convenient for reference to the problem. ANY variable may be substituted for those in the key. In all answers, the expressions on each side of the equations could be reversed. Only one version is listed in the key. For example: 3 + y 8 OR 8 3 + y Solve each problem. You may use the number balance. If you do NOT use the balance to solve it, use it to check your solution. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided 1) Billy Barracuda has found 5 shrimp. His mother wanted him to get 8 for the family s dinner. How many more does he need to find? 5 + y 8 8 y 5 y + 5 8 8 5 y Billy needs to find 3 more shrimp. 2) Billy s mother bought him 12 pencils for school. There are 3 in his backpack and 4 in his desk. The rest are still in the package. How many are left in the package? 12 3 4 g 12 g 4 3 12 g 3 4 12 g 3 + 4 12 7 g 12 4 3 g 12 4 g 3 12 3 g 4 12 g 4 + 3 12 g 7 7 + g 12 g + 7 12 12 g + 3 + 4 Addends may be in any order. If a student writes either of these: 12 3 + 4 g 12 4 + 3 g ask the child to explain his/her reasoning. If the child can explain that he/she added the two known quantities of pencils, and then subtracted the sum from the total number of pencils, he/she understands the situation. Show the child that the correct way to write this would be: 12 (3 + 4) g 12 (4 + 3) g Explain that the parentheses mean do this first. There are 5 left in the package. 3) Billy and his sister, Star, each got $10 for their birthdays. Billy spent $6 and has $4 left. Star spent $7. How much does she have left? 6 + 4 7 s 6 + 4 s 7 6 + 4 7 + s 4 + 6 7 s 4 + 6 s 7 4 + 6 7 + s 10 7 s 7 + s 10 6 + 4 s + 7 10 s 7 s + 7 10 4 + 6 s + 7 Star has $3 left.
4) The 3 Barracuda children each have a collection of beautiful seashells. Star has 5 shells. Billy has 7 shells. Jose has forgotten how many shells he has, but he does know that together the three children have 18 shells. How many shells does Jose have? 18 5 7 j 18 7 j 5 18 5 j 7 13 + j 18 7 + 5 18 - j 18 7 5 j 18 j 7 5 18 j 5 7 j + 13 18 5 + 7 18 j 18 j 13 18 13 j j + 5 + 7 18 Addends may be in any order. If a student writes either of these, 18 7 + 5 j 18 5 + 7 j ask the child to explain his/her reasoning. If the child can explain that he/she added the two known quantities of shells and then subtracted the sum from the total number of shells, he/she understands the situation. Show the child that the correct way to write this would be: 18 (7 + 5) j 18 (5 + 7) j Explain that the parentheses mean do this first. Jose has 6 shells. 5) Billy and his friend, Carol Crab, each have the same number of marbles, some of which are red and some of which are blue. Billy has 3 red and 8 blue. Carol has 2 red. How many blue marbles does Carol have? 3 + 8 2 s 3 + 8 s 2 3 + 8 2 + s 8 + 3 2 s 8 + 3 s 2 8 + 3 2 + s 12 2 s s + 2 12 3 + 8 s + 2 12 s 2 2 + s 12 8 + 3 s + 2 Carol has 9 blue marbles. Extension: Billy s friend, Melanie Mako, ate 7 more fish than Billy ate. Together they ate 17. How many fish did each eat? b + 7 + b 17 b + m 17 (Children must realized that m b + 7 for this one to work.) Students will probably solve this one by guess and check. Accept any explanation that shows understanding of the situation. Billy ate 5 fish. Melanie ate 12 fish.
Name Let Me SHOW What I Know 2 Solve each problem. You may use the number balance. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided Also, draw on the picture of the balance to show how this problem could be solved using an actual number balance. Be sure to label the weights you draw to show what they represent, including writing the variable on the one representing the unknown. 3) Tina Tuna has 6 shiny pebbles. Ike Pike had 3 shiny pebbles. How many shiny pebbles did they have together? Equation: They had pebbles together. 4) Aretha Albacore grew 4 seaweed stalks in her garden. Marvin Manatee has 9 seaweed stalks in his garden. How many more does Marvin have than Aretha? Equation: Marvin has more seaweed stalks than Aretha.
Name KEY Let Me SHOW What I Know 2 Note: Variables used in the key are simply convenient for reference to the problem. ANY variable may be substituted for those in the key. In all answers, the expressions on each side of the equations could be reversed. Only one version is listed in the key. For example: 3 + y 8 OR 8 3 + y Solve each problem. You may use the number balance. Write an equation that represents each situation. Your equation must: contain a variable be written in the boxes provided Also, draw on the picture of the balance to show how this problem could be solved using an actual number balance. Be sure to label the weights you draw to show what they represent, including writing the variable on the one representing the unknown. 1) Tina Tuna has 6 shiny pebbles. Ike Pike had 3 shiny pebbles. How many shiny pebbles did they have together? Tina s pebbles Ike s pebbles pebbles together: p Equation: 3 + 6 p 6 + 3 p They had 9 pebbles together. 2) Aretha Albacore grew 4 seaweed stalks in her garden. Marvin Manatee has 9 seaweed stalks in his garden. How many more does Marvin have than Aretha? Marvin s stalks Aretha s stalks how many more Marvin has than Aretha: x Equation: 4 + x 9 x + 4 9 9 x 4 9 4 x Marvin has 5 more seaweed stalks than Aretha.