## Inspiration

These days, I’ve been a bit obsessed with the double number line. I love using it to teach proportional reasoning – especially percents. Why? A double number line…

- Allows for flexible thinking.
- Expands on a single idea to present a bigger picture.
- Compares percentages
- Builds number sense

Ex. 75% of $36 is $27.

Also,

- A proportion is a section of a double number line.
- The double number line shows students where the percent proportion originates..
- The picture on the right shows 4 different proportions that could answer the question, What is 5% of $36?
- All 4 proportions were pulled from the double number line.

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## The Desmos Activity Builder

Given my current focus on percents, I wanted to create percent challenges where students

- Answer more than one question within the given situation.
- Compare the relationships between percents.
- Visualize the problem as a whole before working on the separate parts.
- Can use a variety of strategies to solve the problem (including a double number line if desired)

This Desmos Activity Builder holds my first set of challenges. Percent – Bar Modeling

**The Activity Builder is broken into 3 parts.**

- Part 1: Learning to use the Percent Bars
- Part 2: The Challenges
- Part 3: Card Sort

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**Part 1: The Percent Bars**

The purpose, of the percent bars, is for students to visually model a given situation. To accommodate varied approaches, the bars stretch to a desired length as well as move about freely. Students acclimate to the percent bars during the first 3 slides.

**Slide 1: **First, I ask students to address the answer. Most students create the this:

I then ask them to change the look of the arrangement. The criteria: The bars must still represent the same percentages, but they’re connected horizontally instead of stacked.

Once the criteria has been set, I start circulating the room. The most common arrangement I see is pictured below. This misconception is the reason I spend a chunk of time teaching students how to use the percent bars. When I come across a student with this arrangement, I’ll start asking questions

- What is the percentage of the blue bar?
**Student: 50%** - Will you drag the blue band under the red?
**Student: Ok**

As the student moves the blue bar, I’ll ask more questions.

- What do you notice?
- What is the percentage of this blue bar?
- How can you fix it?
- What percentage should the green bar be?
- Is it 40% in your picture? Prove it.

**Here are a few successful combinations**

In a whole class format, I’ll highlight 1 or 2 solutions and ask students to prove the percentages for each bar.

**Ex: The red bar starts at 0% and ends at 10%. The green bar starts at the 10% and stops at the 50%. 50% – 10% =40%. The blue bar goes from 50% to 100%. 100% = 5-% = 50%**

**Slide 2:** I asked students to stretch the percent bars to match the given criteria BUT their arrangement had to be different than their shoulder partner’s arrangement.

**Slide 3:** Students assign their own percent values to the bars.

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**Part 2: The Challenges**

Slide 4: We start Challenge 1 together. It’s the only challenge worked on as a whole group. Before students are let loose to work at their own pace or with a partner, I’ll point out 4 features.

### Feature 1: Setting up Visual Model

I learned from experience that many students skim the percentages and start randomly placing marbles. To avoid this behavior and to promote strategic thinking, I required students to first read the directions and set up the percent bars accordingly. To build capacity for flexible thinking, I’ll highlight a few different arrangements.

### **

### Feature 2: Turning on the marbles

Now, that the students have created a visual model of the percentages, I’ll show them how to turn the marbles on and off.

55

### Feature 3: Moving Marbles

If you click on the center of the black point (marble), you can drag it into position.

### 44

### Feature 4: Checking Your Solution

I’ll enter in incorrect values for the red, blue and green marble and explain that if the green check doesn’t pop up, then they must rethink their solution. At this point, students are free to work on their own or with a partner. I do not go over the correct solution to Challenge 1 with the class. I’ve noticed many students choose to start on their own, but start collaborating with their group members quickly.

**Example of correct solution:**

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**Part 3: Card Sort**

The final slide is a card sort. There should be 3 stacks of 5 cards.

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**Teacher Moves while Students are Working**

**Teacher Move 1: Ask questions**

**Challenge 1 sample questions:**

- Does 50% mean a quarter or a half?
- What is the relationship between the red and green percentages?
- If the red and green marbles both represent 25%, should they have to have the same number of marbles?

**Challenge 2 sample questions:**

- What do you know so far?
- How do the green and red percentages compare?
- If you had 2 red marbles, how many green would you have?
- Can you have an odd number of green marbles?
- How many blue bars(10%) fit inside the red bar(30%)?
- If you had 1 blue marble, how many red would you have?
- I noticed you changed your thinking. What inspired your change?

**Challenge 3 sample questions:**

- Tell me what you know?
- Which bar represents Anne?
- Would you drag Anne’s money value into her bar?
- What do you notice?
- How did you know that?
- What percentage amount is 1/4?
- Who has more money? Paul or Anne?

**Challenge 4 sample questions:**

- How do Lilly’s and Chloe’s percentages compare?
- Who has twice as much money as Chloe?
- Tell me something about Lilly and Andrew?
- Did you set up all the percentages?

**Teacher Move 2: Facilitate Collaboration**

- Sara, Jeremy has the same question you just had. Will share with him what you learned?
- (after a few minutes) Jeremy, what did Sara share with you?
- Miguel and Aaron, I see you two are stuck on the same challenge. Will you share with each other what you’ve done so far?
- Group, Crystal has a question. Crystal, will you ask the group your question? I’ll be back to hear about your discussion.
- (If 2 students are working together) Alyssa, Pablo just made a good observation. Will you repeat what he said?

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