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* *Reflection: Intervention and Extension
Building Block Percents - Section 3: Exploration with white boards

Unfortunately, many of my students have learned to rely on procedures when working with fraction concepts. They know tricks and try to remember rules, but many lack number sense with fractions. This fact is a challenge when trying to teach a rigorous Grade 7 curriculum.

This lesson surfaced some concept gaps that I didn't expect. I was really surprised how many students didn't intuitively know that 20% would be 10% + 10%. The Cup Model helped a little, particularly when I asked students how much more they needed to get from 10% to 20%. But, it became very clear, very fast, that the majority of my students were unsure that **10/100 + 10/100** is the same as **20/100**. When I realized this fact I knew that my students:

- Do not interpret a denominator as the name (and size) of the object (unit) being counted
- Do not understand that the numerator is the counting number (or the number of pieces of size denominator)

Today, I ended up teaching a mini-lesson. In it I had students counting unit fractions (see Scaffolding fraction sense) and writing out the denominator name in words. I gave the groups piles of fraction circle pieces and had them count up the one-fourths, for example. When they were done I asked how many one-fourths each group had and wrote it on the board: 3 fourths, 9 fourths, 5 fourths, 4 fourths, etc. I asked two questions to emphasize the definitions of numerator and denominator:

**what is being counted?****which number tells the count?**

Using this information, we were also able to review equivalent mixed number names. I asked questions like, "What is another way to write this same amount?" Using fraction circles was especially helpful for this mini-lesson because it made the whole obvious and concrete.

I had another boy find 10% by dividing the quantity by 10. He was right, but, because I was afraid he was using the wrong 10, I asked him where he got the 10 and he said from the numerator in 10% (10/100). To scaffold &clarify, I asked if he would divide by 30 if he has to find 30% and he confirmed. This told me he didn't understand that the denominator is a divisor and tells how many peices the whole is divided into. I did a brief sidebar with fraction circles with his group relating the denominator to the division and also reminded him of the earlier box diagrams we did.

I could spend a lot more time revisiting fraction concepts with concrete models.

*Intervention and Extension: Processing vs Reasoning with Fractions*

# Building Block Percents

Lesson 17 of 20

## Objective: Students will be able to calculate any whole number percent of a number mentally.

*60 minutes*

The purpose of this lesson is for my students to engage in mental calculations with percentages.

When I first started teaching I noticed that my students did not do percent calculations the way adults do. (Nor do textbooks teach percent calculations in the way that adults do it!) As I watched my students using a pencil and paper algorithm I realized two things:

**The textbook process is complicated for students****In the textbook process there are many opportunities to make mistakes**

In my experience, mental calculations are much faster and generally make more sense to students. When students understand that most percents can be built from 10 Percent they become much more flexible and accurate in their calculations. Moreover, by solving problems mentally students often think about the relationships between the numbers, observing patterns and making generalizations.

#### Resources

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#### Warm up

*30 min*

Students begin today's Warm up when they enter class:

**Angel sells electronics at Best Buy. For every $100 of electronics that he sells he will be paid a $10 bonus (over his regular hourly pay). What percent of sales is his bonus? How much bonus pay will he recieve if he sells $6000 worth of merchandise?**

I expect students to solve this problem in multiple ways. But, I have written the problem in such a way that I anticipate equivalent fractions will be part of the conversation. I know that one misconception my students often have is that if the denominator is a dollar amount (or some other label) that it can't also be the percent. I want my students to understand that **a percent is just a special ratio comparing to a quantity of 100 and that the quantity arises from the context of the problem**.

As always I have students share their answers first in their Math Family Group. Then, we will discuss the problem as a class. As they share multiple methods for finding 10%, I expect my students to make connections between their own strategy and the structure that reasoning with percents gives to a problem (**MP7**). I have included two video examples illustrating my students' thinking about the problem:

- Cristina shares her thinking with the equivalent fraction strategy
- Angelina recognizes the decimal slide

*expand content*

Next, I introduce my Cup Model and I teach students how to "fill the cup." On one side of the cup is the percent full and the other side (opposite ends of the water line) is the quantity in ml. The idea is to build (Scaling up) from 10%.

- I draw the cup with a water line at the top and tell them the capacity when full.
- I tell my students the cup is only 10% full and draw a waterline to represent what that would look like, asking them how much water is in the cup, and I write the amount.
- I draw a new water line and ask how many ml. of water would fill the cup 30%, 50%, etc.
- Students write their answers on their individual white board and raise it up on the count of three so I can assess.

As my students are working out their strategy I circulate to see what they are coming up with. Some students will use repeated addition (10% + 10% + 10%), others will multiply (10% x 3), and others will add parts (10% + 10% + 30% = 50%).

When teaching my students to use this model I start with powers of ten (800 ml., 1200ml.) and stick with percents that are multiples of 10. As my students gain confidence, I start to ask about 5%, 25%, 75%, etc. Most years, the progression moves quickly once a student has shared the halving method for finding 50%.

If there is time to take things a step further: I tell my students how much liquid the cup holds when it is 20% full and I ask them, "How much liquid can the cup hold?"

*expand content*

##### Similar Lessons

Environment: Urban

Environment: Urban

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