Using Strategies in Working Backwards to Solve Multi-Step Word Problems

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SWBAT use strategies to determine an unknown in a multi step word problem.

Big Idea

Students will use a Start,Change and Result strategy, and inverse operations to help them work backwards to find the unknown in a multi-step word problem.

Warm Up!

15 minutes

Hey! What's in the Box?

Whole group warm up: What's in the Box? This number sense game helps students think about strategies we need to use in order to strengthen our number sense in sequencing. You can use a 100's chart that may be posted in your classroom, or print off or use iPads for this website to help them see the patterns. When the game is played, they can't depend on the 100's chart. The idea is to remember the patterns.

Patterns of the 100's chart: Up and Down: + or - 10

Diagonally from the right  corners:  Up and Down : + or - 9. 

Diagonally from the left corners: Up and Down:  + or - 11


Before I gave my students the website, ( or worksheet)  I brought up the What's in the Box notebook page on my Smart Board. I asked them if they could tell me what numbers would be in the boxes? I was trying to get their thinking going and it worked. I was getting them to make sense of the puzzle problem and solve it. ( MP1)

They looked at me like I was nuts. 

I asked them if they thought they could do it if I supplied them with one number. So I wrote the number 36 in the upper right hand corner. 

I still got the deer in the headlights stare. So, I asked them to bring up the website and take a look to find the number 45.  Could they figure out that pattern now? It was easily done this time and we filled in the boxes. I had them put their iPads down and look only at the board.

I asked them at this point if they saw any pattern to the numbers. I expected that they might talk about the numbers 36 and 56 and skip counting by tens, but they noticed that we were skip counting 34 to 36 and the same for 54 to 56. It became a group discussion with everyone chiming in. I could tell they were engaged and wanted to figure out what was going on. 

So, I wrote the number 82 in the next set of squares. I asked them to copy the chart and number in their student notebook ( paper). I asked them to look at their chart and asked what would be above the 82 if there was a box there. They easily produced the number 72. I asked what they thought would be below 82 if there were a box there. And, they easily responded with 92. I asked what numbers we could say we were counting by to figure it out. They said tens.

I wrote the number 109 in one of the boxes, 23 in another and told them to choose one to practice on their own. After a few minutes people were ready to show me their work. Everyone wound up practicing both of them and traded their work with one another to check each other. 

I told them to take this puzzle home and see if their parents could solve it.

This is a fun game that you can do anywhere at anytime and really helps with fluency. 


Lesson: Working Backwards to Solve for an Unknown.

15 minutes

(My Smart Board was not working the day I taught this, so I used my whiteboard for this lesson.)

Whole Class Lesson:

I had all of my students come up and sit together. I started out this lesson with saying " Hey...I am thinking of a number and I want you to figure out what it is...

This caught their attention immediately and I was able to continue. I said: I added 3 and subtracted 12 and my result was 22. What is my number? They started shouting out guesses. Suddenly, my one student who always amazes me said "31!" 

I jumped and they laughed. "How did you get that?" He told me that he was guessing and checking. I think he arrived at it by accident, or the right way, but couldn't explain it. It gave me a great segue into getting them set up to solve problems backwards.

I asked: How about if we think this through and solve it. Let me write it down. I drew an equation as I recanted what I had said, step by step. I stood back and scratched my head, turned and asked...

Do you see the start? Do you see a change? Is there another change? Everyone was on task and understood what I was getting at. I asked out loud why there were two changes in the problem. I got this answer:It's a Multistep Problem! I was pleased that this student had worded it so concisely. 

I then put the "S" above my variable, "C"'s where both should be and then the "R" above the number 22.  I turned and wondered out loud about how we would solve it?

There was stillness for about a minute. They were all perplexed. Even the boy who offered up his correct answer was stumped. 

I wrote the word "inverse operation" on the board. I reviewed fact families out loud and suddenly they knew the meaning of inverse. One girl remembered we had used a water bottle during this discussion in Summer School. I had flipped the water bottle upside down to demonstrate inverse.

So, I told them that if we completely reversed the process, and used inverse operations, we could complete the word problem and know the number I was thinking of.

I began to explain and reverse the process under the equation I had written before. My "Start" was now 22 and I began to reverse the whole process by using the inverse. I spoke that word as I talked. "The inverse of subtracting 12 would be adding 12...our first change this time. Then I would subtract 3 for the second change. My result would be n". 

Then I solved the problem. 

Together we solved several more until they were ready to try a few on their own. Students were called to the board. This boy struggled as I supported him in getting the solution set up. Setting up working backwards. If you notice, the student has not reversed the process. He wants to write the inverse, but in the wrong order in the problem. I fixed this issue after he had all of the equation written out by drawing arrows in the original equation. I noted this as a "common mistake." 

As the lesson progressed, I was able to catch these students in their enthusiasm for solving another problem. Notice, however the two students vying for camera time! Nevertheless, I appreciated that it could be explained. It is evident that he is thinking through his explanation regardless of the antics behind him. What did you get?


Practice and Closure

15 minutes

I told my students that I wanted them to solve three "I am thinking of a number" problems at their desk independently.

I gave them all a template to follow:Lower students were required to solve them and show me before they could write a problem. Mid to Higher students were assigned the money problem #2, to solve and then create three more. They were welcome to discuss with table mates as they worked.

I roved around the classroom and saw varying levels of mastery. I had  interesting discussions with students about money and how many times we think about what is left in our wallet and where did the money go? How do we figure out what we spent when we fear we don't have quite what we thought we should left in our wallet?

As the study time progressed, the majority were able to solve the first problem on the worksheet. Three students really struggled with reversing the process using the S,C,C,R strategy and format. They just wanted to solve it like an algorithm. But, they didn't use the inverse operations. I brought their attention back to their notes and the worksheet and we tried again.

My higher end students had the work done in almost an instant and I assigned them three problems they had to make up and write to have their parents solve. They were to do it on Educreations and save it. A teacher can easily have them do it on paper. 

My two struggling students would need RTI and more monitored practice. 

I chose to close the lesson at this point. I will use the higher students Educreations as part of the RTI for lower students if they turn out well.

If not, I will have to write my own for them and revisit this in the coming week before the test.

My last comment to them was : Have we mastered solving a multi-step problem today? Thumbs up if you completely understand. Sideways if you need more help. Thumbs down if you don't understand yet. Most had thumbs up. Four had sideways thumbs and two had thumbs down. 

This lets me know what I need to address and I will plan on an "after school help" night with more practice. I felt that it was important to let it rest today and not send homework home because the concept will develop with more support. Parents would not be able to support their child at home with homework and I believe that sometimes letting something rest is the best thing to do sometimes.