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* *Reflection: Developing a Conceptual Understanding
Sizing your Own Cup - Section 3: Exploration on white boards

To combat the problems with rushing to procedure when students learn shortcuts try estimation first! When students have figured out the steps to take, even if they understand why they work, they make careless errors and forget to turn on their brain to check if their answer is reasonable. Asking them first to make an estimate may be enough to remind them to check their final answer, especially if you make them write it down. Before finding 40 percent of 20 ml. I had my class think through a quick estimate. I asked "will it be more or less than 20ml? 10ml? 2ml? How do you know?" When they decided it would be a little less than 10 ml. because 40% is a little less than half I made them write it down. 40% of 20ml. is a little less than 10ml. Then, when some of them got 40ml as an answer most of them immediately knew it was wrong and figured out their mistake.

*Estimation is key!*

*Developing a Conceptual Understanding: Estimation is key!*

# Sizing your Own Cup

Lesson 18 of 20

## Objective: Students will be able to mentally calculate percents of numbers.

*54 minutes*

The two most important elements of this lesson are drawing out and emphasizing connections between multiple methods and supporting logical thinking. Students really want to rely on the "decimal slide" as an algorithmic "trick" rather than using the 10% as a building block. I ask students to share & model multiple methods. The biggest struggle students have with this mental model is the logical thinking involved. They have trouble with "If this.... then this...." type of thinking. When finding 30% of 80, they may understand that 30% is three times 10%, but have trouble understanding that it is the 10% value (8 ml.) that they need to multiply by 3 rather than the 10%. When I see students struggling with this I may ask "what is 10% of 80ml?" and "then what do we need to multiply by 3?"Struggling with a common error.wmv

Having the cup model is really helpful for ELL students because as I question I can point to the parts I am referring to. Something else that my ELL students really respond well to is when I try to ask the question in Spanish or Spanglish. "diece porciento par que egual a trenta porciento?" and "Que es trenta porciento ...(as I point to 80ml)" They love when I struggle through it (and completely mangle it) and they get to teach me the right words. They giggle and laugh and tell me how wrong I am doing it. I think that teaching me the words in Spanish helps them develop English vocabulary. It also models for them how to use the visual models. More importantly, it makes them feel more comfortable making an effort and risking looking silly.

#### Resources

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#### Warm up

*20 min*

Instead of giving students a warm up today I displayed problem number 5 from their homework, which was to make their own "cup" percent problem and told them to share their problems with their math "family group" and see if they can solve them. Some of the students skipped that problem, but this gave them a new reason to want to make one up, so they could challenge their peers. I circulated to see the different types of problems students came up with to choose some to share with the whole class.

At each math "family" group I asked students to share their thinking. I continually question students about their work to help students navigate their own thinking. This for me is the best way to help them learn from their mistakes. Rather than correct their mistakes for them I really would rather facilitate their own discovery and correction. It is really important to just keep students explaining and asking them to show you what they did. Then I paraphrase or revoice it to them and ask clarifying questions like: "what is 10% of that number?" or "so, you added 10% and 10%?". Helping students think and assess and revise their work makes them more capable and less reliant on me. Most of the time they figure it out themselves once they make it visible to themselves and hear their words repeated back to them while they refer to the diagram.

Students present own challenges for the class (for example: 10% = 1.5 and 100% = 17) and we explore strategies. (Halving to find 50 percent).

Struggling through it not only helps them make better sense, but it fills them with a sense of pride at their success. Nothing motivates like success!

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#### Exploration on white boards

*30 min*

I took most of the problems for white board practice from the challenges created by students. I displayed the cup model with the given challenge and students work on their individual white boards. Everyone is responsible for their own work, but students can get help from their math "family" groups. The group members must stop and help when one of their members asks for help. I circulate to check for understanding and facilitate learning. All students raise their boards at the same time.

Our first challenge was with a total (100%) cup capacity of 3000 ml. Students found each of the following percents of 300 in order (10%, 20%, 50%, 70%). I asked how they solved the last two and allowed a few students to share multiple methods. Sometimes I call on specific students if I noticed an interesting strategy as they were working. I encouraged students to try out new strategies on the next problems.

Next, I gave them the following information on a cup model: 50% = 80ml. and asked them to find 10%, then 100%, then 60%. Many had found 100% in order to find 10% already. We discussed this as well as other strategies. At this point, some students are filling in even more than I ask for, but some are still struggling.

At this point I let them choose between 10% being 3 or 3.2 and ask them for 20%, 40%, 50%, 100%. The choice gives them more ownership. Because it is shared publically all students are exposed to the extension.

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##### Similar Lessons

Environment: Urban

Environment: Urban

###### Converting Fractions and Decimals

*Favorites(10)*

*Resources(12)*

Environment: Urban

- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Intervention day - Division Remediation
- LESSON 2: Sorting Out Division (day 1 of 5)
- LESSON 3: Sorting Out Division (day 2 of 5)
- LESSON 4: Poster Patterns? (day 3 of 5)
- LESSON 5: Poster Patterns (day 4 of 5)
- LESSON 6: Dominant Traits in Division (day 5 of 5)
- LESSON 7: Fractions in a Box
- LESSON 8: Percents in a Box
- LESSON 9: Is that a Coincidence?
- LESSON 10: Perfracimals 1
- LESSON 11: Perfracimals 2
- LESSON 12: The Cup Half Full (day 1 of 3)
- LESSON 13: Modeling with Box Diagrams on the iPad (day 1 of 2)
- LESSON 14: Modeling with Box Diagrams on the iPad (day 2 of 2)
- LESSON 15: Percent Graffiti
- LESSON 16: The Decimal Slide
- LESSON 17: Building Block Percents
- LESSON 18: Sizing your Own Cup
- LESSON 19: The Cup Half Full
- LESSON 20: Going Mental over Percents