Multiple-Step Problems

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Objective

SWBAT identify the hidden question in a multi-step problem and use it to solve the question being asked of them.

Big Idea

Multi-step problems may require students to solve a hidden question before they can solve the original question.

Opener

5 minutes

In today's lesson, the students learn to answer a multi-step problem by finding the answer to hidden questions. This aligns with 4.OA.A3 because the students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations. I find that students really have a hard time solving multi-step word problems, thus, this is a very necessary skill to be taught.

To get started, I ask the students a question.  "What does multi-step mean?"  I let the students think about the question.  "Share your answer with your neighbor."  By doing this it allows students to share their way of thinking, as well as it may help some students who do not know how to come up with the answer.  I take a few student responses.   Several students responded that multi-step means more than one step.  I was pleased with the responses because this is not our first time hearing about "multi-step" problems.  Periodically, we come across problems that require more than one step in our "board work."  The difference with this lesson is that we address what the students are actually doing to solve the multi-step problems.  I tell the students, "Today, we will learn to find the hidden questions and answer them, in order to answer the original question being asked."

Whole Class Discussion

15 minutes

I call the students to the carpet to discuss the lesson.  The Multiple Step Problems power point is already on the Smartboard.  My students are encouraged to speak out and answer or ask questions during the whole class discussion.  

First, I begin by modeling how to solve a multi-step problem hidden questions.

Problem:

15 of my friends went to the movies last week and paid $8.00 each.  10 of my friends went to the movies this week and paid $8.00 each.  How much did they pay in all?
I explain to the students that in order to find the total, I must first identify the hidden questions in this problem.  The problem is asking me for a total amount that my friends paid at the movies.  Before I can find the total, I must find the amount the friends spent last week and this week. 
The hidden questions:
How much money did my 15 friends spend last week?
How much money did my 10 friends spend this week?

15 x 8 = $120

10 x 8 = $80

Now, we can answer the original question of how much did they pay in all.  We need to add $120 plus $80 to get a total of $200.

I like to model for my students in order to show them what mastery of the skill looks like.
Next, the students help me solve the next problem.
Problem 2:
My mom bakes cupcakes for a bridal shower.  She has 2 boxes of cupcakes that she will deliver.  She has 1 box of cupcakes that has 3 rows with 5 in each row.  She has a second box of cupcakes that has 6 rows with 4 in each row.  How many cupcakes did my mom bake?

What are the hidden questions?  I allow the students to talk with their neighbors to figure out the hidden questions.  This proved to be difficult for the students.  As they discussed, I probed them with questions to help lead them to the hidden questions.  "What are they asking you to find?  What calculations must you make before you can find the total number of cupcakes?"  These types of questions made the students think.  The students knew to multiply the number of rows by the number of cupcakes in the rows, but they could not connect this to a question that they were answering.  We had to refer back to the problem that I modeled for them.  I put it back on the Smartboard to have it in their sights as they thought about their questions.  Finally, they were able to come up with hidden questions that they would need to answer.  

The hidden questions are:

How many cupcakes were in the first box?

How many cupcakes were in the second box?

I ask the students about the operation for the problem.  I ask if there are any clue words to let them know which operation to use.  The students knew to multiply, then add the products.

3 x 5 = 15

6 x 4 = 24

Now, we can add 15 + 24 = 39. There are 39 cupcakes.

 "From these examples, you can see that sometimes you need to find hidden questions in order to answer the original question being asked of you.  You will practice this skill more in groups with your classmates."

Group or Partner Activity

20 minutes

I give the students practice on this skill by letting them work together.  I find that collaborative learning is vital to the success of students.  Students learn from each other by justifying their answers and critiquing the reasoning of others.

For this activity, I put the students in pairs.  I give each group a Group Activity Sheet Multi-Step Problems. The students must decontextualize the problem and represent them symbolically.  The students must work together to solve multiple step word problems.  They must communicate precisely to others within their groups. They must use clear definitions and terminology as they precisely discuss this problem.  The must understand what is being asked of them, make a plan, and persevere to solve the problem (MP1).

The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and must agree upon the answer to the problem.  Because the students must agree upon the answer, this will take discussion, critiquing, and justifying of answers by both students.  As the pairs discuss the problem, they must be precise in their communication within their groups using the appropriate math terminology for this skill.  As I walk around, I am listening for the students to use "talk" that will lead to the answer. I am holding the students accountable for their own learning.  

As they work, I monitor and assess their progression of understanding through questioning. 

1.  What are they asking you to find?

2. Are there any hidden questions that you must solve first?

3.  What operation will you use to solve the problem?

As I walked around the classroom, I heard the students communicate with each other about the assignment.  Before Common Core, I thought that a quiet class working out of the book was the ideal class.  Now, I am amazed at some of the conversation going on in the classroom between the students.  

 My Observations:  

As the students work on this skill, I walk around and monitor.  I hear the students question each other about the hidden questions for the problem.  In the first problem, I hear one pair of students discuss how the teacher collects 15 dollars each time.  They determine that it is a multiplication problem because the number 15 is repeating.  Also, as I monitor, I notice that a pair of students are having a difficult time with the hidden questions. They know that problem 1 is a multiplication problem, and they can solve it.  But, they do not have the hidden questions down.  I ask them, "What is the problem asking you to find?"  The students know that the problem is asking them to find the total money the teacher collected for the dance.  I asked them to explain their first answer to the problem.  They know that the first problem is a multiplication problem of 15 x 6 = 90.  I ask, "What question does this multiplication sentence answer?" The boys think about it for a minute.  To help them further, I ask, "When did the teacher collect the $90?"  The students refer back to the problem and tell me "last week."  "What question does it answer?"  The boys realize that the question is "How much money did the teacher collect for the dance last week."  Questioning is an important tool to help lead students to an understanding.

Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing. 

Closure

15 minutes

To close the lesson, I have one or two students share their answers.  This gives those students who still do not understand another opportunity to learn it.  I like to use my document camera to show the students' work during this time.  Some students do not understand what is being said, but understand clearly when the work is put up for them to see.

I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson.  Students need to see good student samples (Multiple Step Problems), as well as work that may have incorrect information.  More than one student may have had the same misconception.  During the closing of the lesson, all misconceptions that were spotted during the group activity will be addressed whole class.  

Problems Identified in lesson:

What I noticed as I walked around the room and listened in on the groups was that some of the students had a hard time identifying the hidden questions.  They knew which operation to do, which was good.  However, I feel that as 4th graders they should be able to tell me what question is being answered from their calculations.  I feel that this problem exists because students have a hard time with high level thinking.  This is a skill that we must continue to work on this school year.  Over all, I was pleased with how the lesson unfolded.  Some students progressed by the end of the lesson.  We will continue working on identifying hidden questions in multi-step word problems.