Math 6a Unit 3

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In Chapter 3, students begin by focusing on multiple representations of portions, ratios, and equivalence. It then moves into work with integers and signed rational numbers.

Section 3.1 focuses on equivalent fractions and ratios, and introduces the concept of multiplying by the “Giant One,” that is, multiplying both the numerator and denominator of a fraction by the same number to get an equivalent fraction. In this section, students will also examine the connections between fraction, decimal, and percent representations for portions of a whole. Here they look at how the meanings of percents, fractions, and decimals are related, how to represent a quantity in each of these forms, and how to move between these representations. They also connect these ideas to the concept of ratios, which they began looking at with the Trail Mix problem in Lesson 1.1.5. Ratio and proportion are an important focus of this course and this lays the groundwork to connect this big idea to portions and operations with rational numbers.

Section 3.2 begins by having students looking at motion on a number line. Integer expressions are used to represent this motion, with motion to the left being represented by adding a negative number and motion to the right represented by adding a positive number. As they work in this section, students also create their own number lines on which to represent solutions, providing practice with setting intervals and scaling one-dimensional axes in preparation for work with coordinate graphs in Lesson 3.2.4. They work to locate positive and negative numbers on the number line, which leads to finding distance using absolute value. Lastly, they connect this idea of distance to finding the length of a horizontal or vertical line segment on a coordinate graph and plotting points in all 4 quadrants. Students should be familiar with graphing coordinate points in the first quadrant from their work in previous courses.

It is up to you how explicit you want the Mathematical Practices to be in your classroom. If you choose to discuss the Mathematical Practices with your students, now is a good time to stop and do so directly. This can be done as a quick whole group discussion, reflecting on the practices that they have already begun developing through the last two chapters. You might stop and have them do an entry in their Learning Logs related to their use of these practices over the past few weeks. You also might create a class poster and hang it in a prominent place in your room to keep the eight Mathematical Practices front and center in students’ minds.

Please remember, however, that students are not expected to memorize the eight Mathematical Practices. Instead, developing these practices throughout the course will help students gain a more complete view of the mathematics that they are learning. Engaging in mathematics through the practices can also help students recognize the connections between the thinking skills that they learn in math class and those that they use in other areas of their lives.

In the lesson notes for Chapter 3, you are encouraged to revisit many of the standards for mathematical practice. Even if they are not listed, the practices of making sense of problems and persevering to solve them, reasoning abstractly and quantitatively, and attending to precision apply to most every lesson in this course. As you circulate each day, push students to embrace these practices through your questioning.

6.RP.1 - Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.3c - Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

• c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

6.NS.3 - Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4 - Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

6.NS.5 - Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.6 - Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

• a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
• b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.7 - Understand ordering and absolute value of rational numbers.

• a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
• b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
• c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
• d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

6.NS.8 - Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.G.3 - Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

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