This lesson unit is intended to help you assess how well students are able to: solve simple problems involving ratio and direct proportion; choose an appropriate sampling method; and collect discrete data and record them using a frequency table.

Unit 2: Introducing Ratios
Lesson 17: A Fermi Problem

This unit concludes with an opportunity for students to apply the reasoning developed so far to solve an unfamiliar, Fermi-type problem. Students must take a problem that is not well-posed and make assumptions and approximations to simplify the problem (MP4) so that it can be solved, which requires sense making and perseverance (MP1). To understand what the problem entails, students break down larger questions into more-manageable sub-questions. They need to make assumptions, plan an approach, and reason with the mathematics they know.

Unit 3: Unit Rates and Percentages
Lesson 15: Finding This Percent of That

Students have practiced solving three different types of percentage problems (corresponding to finding A, B, or C respectively when A% of B is C). This lesson focuses on finding “A% of B” as efficiently as possible. While the previous lesson used numbers that students could calculate mentally, the numbers in this lesson are purposefully chosen to be difficult for students to calculate mentally or to represent on a double number line diagram, so as to motivate them to find the simplest way to do the calculation by hand.

The third activity hints at work students will do in grade 7, namely finding a constant of proportionality and writing an equation to represent a proportional relationship.

Unit 3: Unit Rates and Percentages
Lesson 16: Finding the Percentage

While students have found percentages with easy numbers before now, in this lesson they will develop a general structure (MP7) that will work for any numbers.

In Module 8, the final module of the year, students extend their understanding of part–whole relationships through the lens of geometry.  As students compose and decompose shapes, they begin to develop an understanding of unit fractions as equal parts of a whole.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In this 35-day Grade 3 module, students extend and deepen second grade practice with "equal shares" to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In this 40-day module, students build on their Grade 3 work with unit fractions as they explore fraction equivalence and extend this understanding to mixed numbers.  This leads to the comparison of fractions and mixed numbers and the representation of both in a variety of models.  Benchmark fractions play an important part in students’ ability to generalize and reason about relative fraction and mixed number sizes.  Students then have the opportunity to apply what they know to be true for whole number operations to the new concepts of fraction and mixed number operations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This 20-day module gives students their first opportunity to explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms.  Utilizing the understanding of fractions developed throughout Module 5, students apply the same reasoning to decimal numbers, building a solid foundation for Grade 5 work with decimal operations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In Module 3, students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions.  Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication.  The module proceeds to fraction by fraction multiplication in both fraction and decimal forms.  An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa.  Students are introduced to the work of division with fractions and decimal fractions.  Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers.  Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.  Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

Overview: This site teaches Ratios and Proportional Relationships to 6th graders through a series of 370 questions and interactive activities aligned to 22 Common Core mathematics skills.

Overview: This site teaches Ratios and Proportions to 6th graders through a series of 588 questions and interactive activities aligned to 23 Common Core mathematics skills.

This site teaches Expressions and Equations to 7th graders through a series of 294 questions and interactive activities aligned to16 Common Core mathematics skills.

Unit 9: Putting It All Together
Lesson 4: How Do We Choose?

This lesson is optional. This is the first of three lessons that explore the mathematics of voting: democratic processes for making decisions. The activities in these lesson build on each other. Doing all of the activities in the three lessons would take more than three class periods—possibly as many as five. It is up to the teacher how much time to spend on this topic. It is not necessary to do the entire set of activities to get some benefit from them, although more connections are made the farther one gets. As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

The activities in this lesson are about voting on issues where there are two choices. Students use proportional reasoning concepts and skills developed in grade 6 to compare voting results of two groups, to determine whether an issue wins an election with a supermajority rule, and discover that a few people can determine the results of an election when very few people vote.

Most of the activities use students’ skills from earlier units to reason about ratios and proportional relationships (MP2) in the context of real-world problems (MP4). While some of the activities do not involve much computation, they all require serious thinking. In many activities, students have to make choices of how to assign votes and justify their methods (MP3).

Most importantly, this lesson addresses topics that are important for citizens in a democracy to understand. Teachers may wish to collaborate with a civics or government teacher to learn how the fictional middle-school situations in this lesson relate to real-world elections.

Unit 2: Introducing Ratios
Lesson 8: How Much for One?

This lesson introduces students to the idea of unit price. Students use the word "per" to refer to the cost of one apple, one pound, one bottle, one ounce, etc., as in “$6 per pound” or “$1.50 per avocado.” The phrase “at this rate” is used to indicate that the ratios of price to quantity are equivalent. (For example, “Pizza costs $1.25 per slice. At this rate, how much for 6 slices?”) They find unit prices in different situations, and notice that unit prices are useful in computing prices for other amounts (MP7).

Students choose whether to draw double number lines or other representations to support their reasoning. They continue to use precision in stating the units that go with the numbers in a ratio in both verbal statements and diagrams (MP6).

Note that students are not expected to use or understand the term "unit rate" in this lesson.

Unit 3: Unit Rates and Percentages
Lesson 6: Interpreting Rates

In previous lessons students have calculated and worked with rates per 1. The purpose of this lesson is to introduce the two unit rates, a/b and b/a, associated with a ratio a:b. Each unit rate tells us how many of one quantity in the ratio there is per unit of the other quantity. An important goal is to give students the opportunity to see that both unit rates describe the same situation, but that one or the other might be preferable for answering a given question about the situation. Another goal is for students to recognize that they can just divide one number in a ratio by another to find a unit rate, rather than using a table or another representation as an intermediate step. The development of such fluency begins in this section and continues over time. In the Cooking Oatmeal activity, students have explicit opportunities to justify their reasoning and critique the reasoning of others (MP3).

Unit 2: Introducing Ratios
Lesson 6: Introducing Double Number Line Diagrams

This lesson introduces the double number line diagram, a useful, efficient, and sophisticated tool for reasoning about equivalent ratios.

The lines in a double number line diagram are similar to the number lines students have seen in earlier grades in that:

Numbers correspond to distances on the line (so that the distance between, say, 0 and 12 is three times the distance between 0 and 4);
We can choose what scale to use (e.g., whether each interval represents 1 unit, 2 units, 5 units, etc.);
The lines can be extended as needed.
In a double number line diagram we use two parallel number lines—one line for each quantity in the ratio—and choose a scale on each line so equivalent ratios line up vertically.

For example, if the ratio of number of eggs to cups of milk in a recipe is 4 to 1, we can draw a number line for the number of eggs and one for the cups of milk. On the number lines, the quantity of 4 for the number of eggs and the 1 for cups of milk would line up vertically, as would 8 eggs and 2 cups of milk, and so on.

Because they represent quantities with length on a number line rather than with counts of objects, double number lines are both more abstract and more general than discrete diagrams. Later in this unit, students will learn an even more abstract representation of equivalent ratios—the table of values. Connecting the concrete to the abstract helps students connect quantitative reasoning to abstract reasoning (MP2). Though some activities are designed to hone students’ facility with particular representations, students should continue to have autonomy in choosing representations to solve problems (MP5), as long as they can explain their meaning (MP3).

Unit 2: Introducing Ratios
Lesson 1: Introducing Ratios and Ratio Language

In this lesson, students use collections of objects to make sense of and use ratio language. Students see that there are several different ways to describe a situation using ratio language. For example, if we have 12 squares and 4 circles, we can say the ratio of squares to circles is 12:4 and the ratio of circles to squares is 4 to 12. We may also see a structure that prompts us to regroup them and say that there are 6 squares for every 2 circles, or 3 squares for every one circle (MP7).

Expressing associations of quantities in a context—as students will be doing in this lesson—requires students to use ratio language with care (MP6). Making groups of physical objects that correspond with “for every” language is a concrete way for students to make sense of the problem (MP1).

It is important that in this first lesson students have physical objects they can move around. Later, they will draw diagrams that reflect the same structures and learn to reason with and interpret abstract representations like double number line diagrams and tables. Working with objects that can be physically rearranged in the beginning of the unit can help students make sense of increasingly abstract representations they will encounter as the unit progresses. Students will continue to develop ratio language throughout the unit and will learn about equivalent ratios in a future lesson.

Unit 3: Unit Rates and Percentages
Lesson 3: Measuring with Different-Sized Units

This lesson develops students’ familiarity with standard units of length, volume, weight, and mass through the tactile experiences of measuring objects. The main idea is that it takes more of a smaller unit and less of a larger unit to measure the same quantity. This idea is an important foundation for converting units of measurement using ratio reasoning in the next lesson (MP7).

In this activity, students will focus on subtracting fractions, starting with single-digit denominators and working up.