Equivalent Expressions Using Multiplication
Math Direct Instruction Lesson Plan
Grade Level/Subject: 5th Grade/Math Topic: Multiplication
Rationale:
Students in 5th grade need to be able to reason about equivalent expression in multiplication. Students need to generate equivalent multiplication expressions, use story contexts and representations to support explanations of the relationship between equivalent expressions, and develop arguments about how to generate equivalent expression in multiplication. Being able to generate and explain equivalent expressions is a skill that students will need to succeed in mathematics.
Common Core/Essential Standards Reference:
5.NBT.B.5: Fluently multiply multi-digit whole numbers using the standard algorithm.
Behavioral Objective:
Students will be able to generate equivalent multiplication expressions by doubling (or tripling) one factor and dividing the other by 2 (or 3), use story contexts and representations to support explanations of the relationship between equivalent expressions, and develop arguments about how to generate equivalent expressions in multiplication. To be successful, students will be able to accurately answer 8 out of 10 questions on the final worksheet.
Prerequisite Knowledge and Skills:
Knowledge: Number recognition, recognition of number equality, how to compare numbers, story problem creation
Skills: Multiplication facts
Materials/Resources:
Investigations in Number, Data, and Space, Unit 7, Multiplication and Division 2, pp. 26-36
Pages 1-5, Unit 7, Investigations Student book
PPT slide show
Content and Strategies
_______________________________________________________________________________________________
Focus/Review:
Ask students “How do you know if two equations are equal?” and “If you are creating two equivalent equations, is there a way to do that without solving them first?” Explain to students that we are going to use what we already know about multiplication and division to create equivalent equations without having to solve them.
Objective (as stated for students):
Today we’re going to use story problems and representations to explain the relationship between equivalent expressions. We are going to create equivalent multiplication expressions by doubling (or tripling) one factor and dividing the other.
Teacher Input:
Write the following equation on the board: 8 x 12 = 4 x 24
Read the following story problem to students:
Eight authors wrote books that are each 12 chapters long. The authors decided to pair up and combine their books. Now there are 4 books that are 24 chapters long.
Tell students that in order to create equivalent multiplication expressions the factors must change in a certain way in order for them to be equivalent. Each side of the = needs to have the same product in order for the equation to be true. When you are comparing equations, their products need to be equal.
Guided Practice:
(Start PPT slideshow)
Write 6 x 9 = 3 x 18 on the board and ask:
· Is this true?
· What is the relationship between the two sides of the equations?
· Is there a way you can tell without multiplying?
Have students work with partners to share ideas. Teacher will listen for the doubling and halving relationship. Once students have had time to discuss, explain that they have solved equivalent problems in which the number were doubled and halved. One factor was doubled and one was halved in order to make the same product.
Ask students to think of a story problem that shows 6 x 9 = 3 x 18. Tell them to start with a story for 6 x 9 then change the situation to 3 x 18. Have students work with a partner to come up with a story problem that makes sense.
Have several students share stories.
Ask students:
· Do you think this doubling and halving relationship in multiplication is true for other numbers?
· Think about your story problems. What if you changed the numbers?
Guide students to provide a clear statement of the conjecture about doubling and halving in multiplication. Have students come up with a rule for doubling and halving. Ask students “What is the idea, the conjecture, we’re trying to prove?” If needed, have students start their sentence: “If you are multiplying two numbers….” Have students work with a partner to show whether the doubling and halving rule will always work.
Have students create a representation that shows their thinking about the rule. Tell them it can be a picture, a diagram, or a model. They can use the story problems they created earlier. Students may work with partners. Have students use page 1, unit 7 to record their work. Teacher will visit pairs/groups and ask:
· Where are the groups in your picture?
· Where can I see the doubling and halving?
· How does your picture show that this will always work?
· What if I changed these numbers to a __ and a ___? Would your picture still work?
Once students have had time to create their representations, ask them to share with the class. Students will come to the board and recreate their representations and explain their thinking.
Have students complete page 2, unit 7 independently to reinforce concept.
Write the following problem on the board: 2 x 10 = 4 x ___. Give students a few minutes to fill in the missing number to make the equation true. Have students think of a story problem for this equation, ask a student to share their problem. (Allow 1-2 minutes for this problem)
Read the following story problem to the class:
Imagine that there is one clown on stage juggling 15 apples. Two more clowns come on stage. If the clowns always split up the apples evenly, how many apples will each clown be juggling now? Then ask: How do you know? Allow students to discuss with a partner and figure out an equivalent equation.
Write 1 x 15 = 3 x 5 on the board. Tell students that this time, instead of doubling the number of clowns, it was tripled. Ask: What happened to the number of apples for each clown? Allow students to discuss with a partner.
Write 2 x 9 = 6 x ___ on the board. Instruct students to create a story problem and a representation of this equivalent equation. Students may use a picture, diagram, or model. Once students finish with their representation, have them try tripling and thirding other numbers. Ask:
· Does it always work to triple on factor and divide the other in thirds?
· How do you know?
Have students use page 3, unit 7 from their Investigations book to record their work.
Once students have had time to create their representations, ask them to share with the class. Students will come to the board and recreate their representations and explain their thinking.
Have students complete page 4, unit 7 independently to reinforce concept.
Independent Practice:
Students will independently work on pages 2 and 4 from their Investigations math book during the lesson. While students are working on pages 2 and 4, the teacher will walk around the room checking for understanding. Students will need to be able to explain their representations they have created. Students will also independently complete page 5 if time permits, if not, students will complete this page as homework.
Closure:
Ask students what the rule for doubling and halving is. Ask if this is true for all numbers. Ask if it’s the same principle for tripling and thirding.
Evaluation:
Students will be evaluated on their ability to correctly create a representation that shows their thinking and that their equation is true. Also, students will need to accurately answer 8 out of 10 questions on page 5 from their Investigations book.
Plans for Individual Differences:
For ELL students, I will make a reference chart that explains what doubling, halving, tripling, and thirding mean. Students are asked to understand these concepts throughout this lesson and the chart will allow them to see the concepts.
Grade Level/Subject: 5th Grade/Math Topic: Multiplication
Rationale:
Students in 5th grade need to be able to reason about equivalent expression in multiplication. Students need to generate equivalent multiplication expressions, use story contexts and representations to support explanations of the relationship between equivalent expressions, and develop arguments about how to generate equivalent expression in multiplication. Being able to generate and explain equivalent expressions is a skill that students will need to succeed in mathematics.
Common Core/Essential Standards Reference:
5.NBT.B.5: Fluently multiply multi-digit whole numbers using the standard algorithm.
Behavioral Objective:
Students will be able to generate equivalent multiplication expressions by doubling (or tripling) one factor and dividing the other by 2 (or 3), use story contexts and representations to support explanations of the relationship between equivalent expressions, and develop arguments about how to generate equivalent expressions in multiplication. To be successful, students will be able to accurately answer 8 out of 10 questions on the final worksheet.
Prerequisite Knowledge and Skills:
Knowledge: Number recognition, recognition of number equality, how to compare numbers, story problem creation
Skills: Multiplication facts
Materials/Resources:
Investigations in Number, Data, and Space, Unit 7, Multiplication and Division 2, pp. 26-36
Pages 1-5, Unit 7, Investigations Student book
PPT slide show
Content and Strategies
_______________________________________________________________________________________________
Focus/Review:
Ask students “How do you know if two equations are equal?” and “If you are creating two equivalent equations, is there a way to do that without solving them first?” Explain to students that we are going to use what we already know about multiplication and division to create equivalent equations without having to solve them.
Objective (as stated for students):
Today we’re going to use story problems and representations to explain the relationship between equivalent expressions. We are going to create equivalent multiplication expressions by doubling (or tripling) one factor and dividing the other.
Teacher Input:
Write the following equation on the board: 8 x 12 = 4 x 24
Read the following story problem to students:
Eight authors wrote books that are each 12 chapters long. The authors decided to pair up and combine their books. Now there are 4 books that are 24 chapters long.
Tell students that in order to create equivalent multiplication expressions the factors must change in a certain way in order for them to be equivalent. Each side of the = needs to have the same product in order for the equation to be true. When you are comparing equations, their products need to be equal.
Guided Practice:
(Start PPT slideshow)
Write 6 x 9 = 3 x 18 on the board and ask:
· Is this true?
· What is the relationship between the two sides of the equations?
· Is there a way you can tell without multiplying?
Have students work with partners to share ideas. Teacher will listen for the doubling and halving relationship. Once students have had time to discuss, explain that they have solved equivalent problems in which the number were doubled and halved. One factor was doubled and one was halved in order to make the same product.
Ask students to think of a story problem that shows 6 x 9 = 3 x 18. Tell them to start with a story for 6 x 9 then change the situation to 3 x 18. Have students work with a partner to come up with a story problem that makes sense.
Have several students share stories.
Ask students:
· Do you think this doubling and halving relationship in multiplication is true for other numbers?
· Think about your story problems. What if you changed the numbers?
Guide students to provide a clear statement of the conjecture about doubling and halving in multiplication. Have students come up with a rule for doubling and halving. Ask students “What is the idea, the conjecture, we’re trying to prove?” If needed, have students start their sentence: “If you are multiplying two numbers….” Have students work with a partner to show whether the doubling and halving rule will always work.
Have students create a representation that shows their thinking about the rule. Tell them it can be a picture, a diagram, or a model. They can use the story problems they created earlier. Students may work with partners. Have students use page 1, unit 7 to record their work. Teacher will visit pairs/groups and ask:
· Where are the groups in your picture?
· Where can I see the doubling and halving?
· How does your picture show that this will always work?
· What if I changed these numbers to a __ and a ___? Would your picture still work?
Once students have had time to create their representations, ask them to share with the class. Students will come to the board and recreate their representations and explain their thinking.
Have students complete page 2, unit 7 independently to reinforce concept.
Write the following problem on the board: 2 x 10 = 4 x ___. Give students a few minutes to fill in the missing number to make the equation true. Have students think of a story problem for this equation, ask a student to share their problem. (Allow 1-2 minutes for this problem)
Read the following story problem to the class:
Imagine that there is one clown on stage juggling 15 apples. Two more clowns come on stage. If the clowns always split up the apples evenly, how many apples will each clown be juggling now? Then ask: How do you know? Allow students to discuss with a partner and figure out an equivalent equation.
Write 1 x 15 = 3 x 5 on the board. Tell students that this time, instead of doubling the number of clowns, it was tripled. Ask: What happened to the number of apples for each clown? Allow students to discuss with a partner.
Write 2 x 9 = 6 x ___ on the board. Instruct students to create a story problem and a representation of this equivalent equation. Students may use a picture, diagram, or model. Once students finish with their representation, have them try tripling and thirding other numbers. Ask:
· Does it always work to triple on factor and divide the other in thirds?
· How do you know?
Have students use page 3, unit 7 from their Investigations book to record their work.
Once students have had time to create their representations, ask them to share with the class. Students will come to the board and recreate their representations and explain their thinking.
Have students complete page 4, unit 7 independently to reinforce concept.
Independent Practice:
Students will independently work on pages 2 and 4 from their Investigations math book during the lesson. While students are working on pages 2 and 4, the teacher will walk around the room checking for understanding. Students will need to be able to explain their representations they have created. Students will also independently complete page 5 if time permits, if not, students will complete this page as homework.
Closure:
Ask students what the rule for doubling and halving is. Ask if this is true for all numbers. Ask if it’s the same principle for tripling and thirding.
Evaluation:
Students will be evaluated on their ability to correctly create a representation that shows their thinking and that their equation is true. Also, students will need to accurately answer 8 out of 10 questions on page 5 from their Investigations book.
Plans for Individual Differences:
For ELL students, I will make a reference chart that explains what doubling, halving, tripling, and thirding mean. Students are asked to understand these concepts throughout this lesson and the chart will allow them to see the concepts.
Doubling
Halving Tripling Thirding |
Multiply by 2
Divide by 2 Multiply by 3 Divide by 3 |
X 2
÷ 2 X 3 ÷ 3 |
Lesson Reflection
I spoke with my cooperating teacher about what lesson she wanted me to teach and we decided that I would wait until the students began a new unit. They were going to be covering new strategies in determining if equations were equivalent. My teacher wanted me to write a lesson that covered doubling one factor/halving the other factor and tripling/thirding, even though I probably wouldn’t get to the tripling/thirding portion. My teacher thought that if the students caught on quickly, I may be able to start on tripling/thirding. By the time we came to the second part of the lesson, there was only 5 minutes left in the math block and we decided that the students would learn the rest the next day. Unfortunately, the day I taught my lesson was my last day in the classroom so I wasn’t able to teach the second portion.
I thought the lesson went very well. Once we started on guided practice, several students saw the relationship between the two sides of the equation right away and were able to explain their thinking to the class. The students were familiar with story problems so they were able to quickly come up with some to explain the expressions. There were a few students that struggled in the beginning because I asked them not to multiply to see if they could find out if the expressions were equal. However, once we started discussing the relationship, they caught on.
I had students work in pairs at first and then work with their table groups. This worked out really well. The students understood that I was going to let them talk during the lesson, so they were more focused when I was speaking. The students were respectful of each other and their opinions.
The only part of the lesson when the students had issues was when I asked them to come up with a rule/conjecture about doubling and halving in multiplication. I had them try to come up with a rule with their table groups and when I visited each group, I had to go into more detail. When I talked about it to the whole group, I don’t think I was very clear on what I wanted them to do. However, when I walked around the room, I clarified the instructions to each group. Next time, I’ll need to be clearer in what I’m expecting and I would probably bring the whole group back together to explain one more time instead of 6 separate times.
Planning the lesson was made easier because my cooperating teacher gave me the Investigations book to teach from. The problems were already listed as well as what the students should be doing and the teacher should be saying. It listed the concepts that you want the students to understand as well as provided examples.
I thought the lesson went very well. Once we started on guided practice, several students saw the relationship between the two sides of the equation right away and were able to explain their thinking to the class. The students were familiar with story problems so they were able to quickly come up with some to explain the expressions. There were a few students that struggled in the beginning because I asked them not to multiply to see if they could find out if the expressions were equal. However, once we started discussing the relationship, they caught on.
I had students work in pairs at first and then work with their table groups. This worked out really well. The students understood that I was going to let them talk during the lesson, so they were more focused when I was speaking. The students were respectful of each other and their opinions.
The only part of the lesson when the students had issues was when I asked them to come up with a rule/conjecture about doubling and halving in multiplication. I had them try to come up with a rule with their table groups and when I visited each group, I had to go into more detail. When I talked about it to the whole group, I don’t think I was very clear on what I wanted them to do. However, when I walked around the room, I clarified the instructions to each group. Next time, I’ll need to be clearer in what I’m expecting and I would probably bring the whole group back together to explain one more time instead of 6 separate times.
Planning the lesson was made easier because my cooperating teacher gave me the Investigations book to teach from. The problems were already listed as well as what the students should be doing and the teacher should be saying. It listed the concepts that you want the students to understand as well as provided examples.