Proportions, Rational Numbers, Equations & Chance
Objectives
- D05.S1.07.01 Use proportional relationships, including unit rates, scale, and percentages, to solve multi-step problems in real contexts.
- D05.S1.07.02 Perform all four operations with rational numbers such as fractions, decimals, and integers, and explain each step of the reasoning.
- D05.S2.07.01 Write and solve two-step equations and inequalities to represent and solve real problems.
- D05.S2.07.02 Identify proportional relationships in tables, graphs, and equations, and express them in the form y = kx.
- D05.S3.07.01 Construct and describe triangles and other shapes from given conditions, and tell when such a shape is impossible.
- D05.S3.07.02 Solve problems with scale drawings and circles, using the relationship between circumference and diameter.
- D05.S4.07.01 Use random sampling to draw conclusions about a population and recognize bias in how data is collected.
- D05.S4.07.02 Investigate chance processes and compute probabilities of simple and compound events.
Essential questions
- How do proportional relationships, rational numbers, and percentages let me reason about real situations?
- How do equations and inequalities turn real problems into solvable form, and how do I read y = kx?
- How do sampling, bias, and probability let me draw careful conclusions from data and chance?
Big ideas
- Proportional relationships, unit rates, scale, and percentages model real shares, prices, and sizes.
- Two-step equations and inequalities and the form y = kx turn real problems into reasoned solutions.
- Random sampling reveals a population while bias distorts it, and probability measures chance in simple and compound events.
Lessons
- 1 Unit Rates in Real Life 50 minutes
- 2 Percentages: Discounts, Tax, and Tips 50 minutes
- 3 Multi-Step Proportion Problems 50 minutes
- 4 Operating with Fractions and Decimals 50 minutes
- 5 Operating with Integers 50 minutes
- 6 Writing and Solving Two-Step Equations 50 minutes
- 7 Two-Step Inequalities 50 minutes
- 8 Proportional Relationships in Tables and Graphs 50 minutes
- 9 Proportional Relationships as y = kx 50 minutes
- 10 Building Triangles: Possible and Impossible 50 minutes
- 11 Scale Drawings 50 minutes
- 12 Circles and the Number π 50 minutes
- 13 Random Sampling and Bias 50 minutes
- 14 Probability of Simple and Compound Events 50 minutes
Assessment plan
Observation of proportional and rational-number reasoning and of equation-building; a multi-step proportion problem, a two-step equation-and-inequality set, a circle and scale-drawing task, and a sampling-and-probability investigation as portfolio artifacts.
Unit 5 — Proportions, Rational Numbers, Equations & Chance
Overview
This unit grows from Grade 6’s ratios, one-step equations, and data summary into the full working of algebra and chance. Learners use proportional relationships — unit rates, scale, and percentages — to solve multi-step problems in real contexts, and perform all four operations with rational numbers, explaining each step of the reasoning. They write and solve two-step equations and inequalities to represent and solve real problems, and identify proportional relationships in tables, graphs, and equations, expressing them in the form y = kx. They construct and describe triangles and other shapes from given conditions and tell when such a shape is impossible, solve problems with scale drawings and circles using the relationship between circumference and diameter, use random sampling to draw conclusions about a population and recognize bias in how data is collected, and investigate chance processes and compute probabilities of simple and compound events. Mathematics remains sense-making with real things, drawn from many traditions, for everyone.
Essential questions
- How do proportional relationships, rational numbers, and percentages let me reason about real situations?
- How do equations and inequalities turn real problems into solvable form, and how do I read y = kx?
- How do sampling, bias, and probability let me draw careful conclusions from data and chance?
Lessons
| Lesson | Title | Standards |
|---|---|---|
| L.07.005.01 | Unit Rates in Real Life | D05.S1.07.01 |
| L.07.005.02 | Percentages: Discounts, Tax, and Tips | D05.S1.07.01 |
| L.07.005.03 | Multi-Step Proportion Problems | D05.S1.07.01 |
| L.07.005.04 | Operating with Fractions and Decimals | D05.S1.07.02 |
| L.07.005.05 | Operating with Integers | D05.S1.07.02 |
| L.07.005.06 | Writing and Solving Two-Step Equations | D05.S2.07.01 |
| L.07.005.07 | Two-Step Inequalities | D05.S2.07.01 |
| L.07.005.08 | Proportional Relationships in Tables and Graphs | D05.S2.07.02 |
| L.07.005.09 | Proportional Relationships as y = kx | D05.S2.07.02 |
| L.07.005.10 | Building Triangles: Possible and Impossible | D05.S3.07.01 |
| L.07.005.11 | Scale Drawings | D05.S3.07.02 |
| L.07.005.12 | Circles and the Number π | D05.S3.07.02 |
| L.07.005.13 | Random Sampling and Bias | D05.S4.07.01 |
| L.07.005.14 | Probability of Simple and Compound Events | D05.S4.07.02 |