Shopping Comparison Problems
Lesson 8 of 8
Objective: SWBAT multiply to solve word problems involving multiplicative comparison.
Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to explore the distributive property. I began by explaining the Distributive Property Poster: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example: a x (b + c) = (a x b) + (a x c).
Modeling the Distributive Property: 2 x (5 + 3)
I wrote this task on the board: Teacher Model, 2 x (5 +3). I reminded students: We always solve whatever is inside the parenthesis first. So what's 5 + 3? Students responded, "8!"What's 8 x 2? Students responded, "16!" What would happen if we distributed the two by multiplying the 2 by the 5 and the 2 by the 3? I then wrote: (2 x 5) + (2 x 3) on the board. Students did the same on their own white boards: Student White Board 2 x (5 + 3). Students began solving this equation on their own. In no time, I heard expressions of astonishment, "It worked!" "Wow! We go the same answer!" I then responded: I wonder if this happens every time!
Students Testing the Distributive Property
Next, I asked students to test the distributive property using their own numbers. Here are a few examples:
Some students even tried distributing across three addends!
As students completed each task, I asked them to share their findings with others. We also shared strategies as a class. I modeled students' thinking on the board while the students used math words to explain how they used the distributive property. I then asked: Is the distributive property always true? Does it always work? Through investigating this property on their own, most students were convinced that they could trust the distributive property.
Goal & Introduction
To begin, I invited students bring their whiteboards and sit on the front carpet, closer to the board. I introduced today's goal: I can multiply to solve word problems involving multiplicative comparison. I wrote the following on the board: Abby and Brynn went shopping. Abby spent __ times as much as Brynn. How much did Abby spend if Brynn spent $____?
Then I explained: Let's say that Brynn and Abby (two students in our class) went shopping. Abby is in the mood to spend money! (Students giggled!) In fact, whatever Brynn spends, Abby always likes to spend ___ times as much. Let's say that Brynn wants to save her money.
This problem is a multiplicative comparison problem. Turn and talk: What key words helped you know that there was a comparison happening here? Here's the class discussion the followed: Comparison Problem Discussion.
I continued on by drawing the following process grid on the board: Process Grid. I wanted to just focus on the first two columns to begin with. Here's what it they will look like when we are done: Comparison Statement & Bar Diagram.
I filled in the first blank of the Problem with the number 2 and left the second blank empty. I explained: Let's say we represent the amount that Abby spent with a and the amount that Brynn spent with b. Let's also say that Abby spent two times as much as Brynn. How might we write a and b in an equation to show this comparison? I began by writing... "2 x..." I then asked: What will we multiply by two... would we multiply the amount Abby spent by 2 or the amount Brynn spent by 2? Students immediately responded, "Brynn because Abby spent 2 x as much as Brynn." We then came up with 2 x b =a. It wasn't long until a student pointed out that b also equals 1/2 of a: 2 x b = a or 1:2 x a = b.
We moved on to the bar diagram. This is a model that we've used several times in the past to model word problems. How might we represent this equation using a bar diagram? Students began drawing and discussing bar diagrams on their white boards. Eventually, we all came up with: b + b = a.
Next, I erased the 2 in the problem and replaced it with 5. We then discussed how this would change the comparison statement to 5 x b =a. Again, students modeled the bar diagram.
Finally, I erased the 5 in the problem and replaced it with 10. We discussed how this would change the comparison statement to 10 x b = a. Again, students modeled the bar diagram.
What Happens if b = $5?
It was now time to complete the second part of the process grid: Value of b and a. I explained: Now, I'm going to finally tell you how much Brynn spent! When Abby and Brynn went shopping, Brynn only spent $5. Now that we know the value of b, could we find the value of a? Let's rewrite the comparison statement and replace the b with 5 (2 x $5 =$10). If Abby spent twice as much as Brynn, how much did Abby spend? In other words, what's the value of a? Students caught on quickly, "A = $10!" Does that make sense? If Abby spent twice as much as Brynn? A student responded, "Yes because $10 is twice as much as $5!"
I knew students were ready to continue practicing with their partners!
Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students loved being able to develop a "game plan" with their partners!
I asked students to return to their desks and I passed out a copy of the Winter Gear to each pair of students. To help students become engaged in the next problem solving task, I created and presented a Powerpoint called Multiplicative Comparison. I explained: The Smith Family needs your help today! Just like Abby and Brynn, the Smith family is going shopping! They need help purchasing winter wear for their family at the Family Sports Store. We're going to solve one problem altogether and then you'll have three more problems to solve with your partner.
I showed students the First Problem: Snow pants cost 3 times as much as gloves. If the Smith family buys 4 sets of snow pants and five sets of gloves, how much will the snow pants and gloves be altogether? We then discussed the two steps in this problem: #1 Find out how much the snow pants are and #2 Find the total cost of 4 snow pants and 3 gloves.
We began by writing a comparison statement: s = 3 x g. We discussed how we could make our work more precise by writing s = snow pants and g = gloves. We also made a bar diagram to show that 3 x g = s. We then used the Winter Gear resource to determine the cost of gloves, $19. I purposefully encourage students to use variables and a bar digram to help engage students in Math Practice 2: Reason abstractly and quantitatively.
Next, we used the standard algorithm to multiply 19 x 3 in order to find the cost of the snow pants, $57. I modeled how to double check our work by using a second strategy (decomposing) to encourage students to engage in Math Practice 3: Construct viable arguments. I asked: Is $57 the answer to the problem. Some students said, "Yes!" Others said, "No, there's a second step.
We then multiplied the cost of snow pants, $57, by 4 and the cost of gloves, $19, by 5 to arrive at the total, $323.
I passed out a copy of the Smith Family Problems to each student. As we have done many times in the past, students glued a problem down on each page in their journals and got right to work on the first problem!
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by asking guiding questions (listed below). I also focused on not giving away answering or correcting student mistakes.
- How can you make your work more precise?
- What does the "c" represent?
- What are you going to do next?
- Do you know the value of c or b?
- Can you show me how you found that?
- Can you explain your thinking?
- Does it make sense to ______?
- What does make sense?
During this conference, a student explains how she used variables and bar diagrams to represent the Boots & Coats Problem.
Here, a student explains how he focused on Finding Mistakes. I was so proud of him for discovering his mistake on his own!
Many students were able to complete all four problems, however, more importantly, all students were engaged in problem solving and persevering! In addition, all students successfully understood multiplicative comparison problems. Here are examples of finished work: