Multiplication
Groups, Arrays, and Facts
About this book
Compare quantities fairly: right order, right units, right rate.
Ratios and Unit Rates helps learners compare quantities with care. They write ratios in the order a question names, tell part-to-part from part-to-whole, and build equivalent ratios with tables, tape diagrams and double number lines. They find unit rates such as cost per item and kilometers per hour, and they decide when a bigger or a smaller rate is better. The last module tests whether a fixed ratio fits a situation at all. Every module follows the same steps, from a worked example to independent practice and review. An explanatory answer key at the back shows reasons, common errors and reteaching cues.
- Readiness check with routes to earlier skills
- Four modules, each with the same seven steps
- Worked examples, guided and independent practice
- Cumulative review, exit check and a short retry
- Answer key with reasons and common errors
For learners in grades 6–8 who can multiply, divide, read simple tables and use fractions.
Look inside
What this book covers
Covered
- This book is about ratios and unit rates.
- You will write ratios in the right order, make equivalent ratios, find unit rates, and decide when a fixed ratio really fits a situation.
- Extra pages after p. 52 read rates from graphs, convert units and use scale drawings.
Not covered
Not in this book: percents (see Percentages in Everyday Problems) and writing equations for proportions (see Expressions and One-Variable Equations and Linear Equations and Graphs).
Before you start
You should already know the skills in Multiplication: Groups, Arrays, and Facts, Division: Equal Groups, Sharing, and Remainders and Fractions: Parts, Sets, and Number Lines.
Things to have
Use a calculator to check work, not replace it.
Notice
Pictures in this book are simplified learning models. Check answers with the explanations at the back of the book.
The steps in this book
- Step 1: Get Ready
Check the skills you need before you begin. If something feels tricky, an adult can help you start in the right place.
- Step 2: Learn It
Read a short, clear explanation of the new idea and the words that go with it.
- Step 3: See It
Watch the idea being used, one step at a time, in a worked example.
- Step 4: Try It Together
Practice with prompts and help cues. The help gets smaller as you get stronger.
- Step 5: On Your Own
Practice on your own with different kinds of questions, and explain your thinking.
- Step 6: Look Back
Look back at earlier learning so it stays strong, then check where to go next.
- Step 7: Check and Fix
Check your answers. Each answer explains why, names a common mistake, and shows how to fix it.
What is in the book
The modules
- Module 1: Ratios describe relationshipsPages 5–14
- Module 2: Equivalent ratio representationsPages 15–24
- Module 3: Unit rates and useful unitsPages 25–34
- Module 4: Choose a proportional strategyPages 35–44
Contents
- Readiness check, pp. 3–4. Before Module 1. Its table tells you where to start.
- Modules, pp. 5–44
- Checkpoints, pp. 14, 24, 34, 44. The last page of each module.
- Mixed review, pp. 45–48. Questions from across the book, mixed together.
- Exit check, pp. 49–50. A last check across the book.
- Reflect and repair, p. 51
- Try again, p. 52. Fresh questions like those in the exit check.
- Further practice, pp. 53–55. More pages of step 5, On Your Own.
- Extra practice, pp. 56–58
- Answers and explanations, pp. 59–68. Notes in this part are headed “Why”, “Common mistake” and “Help”.
Other books it mentions
Also mentioned
Percentages in Everyday Problems
Parts, wholes, change, and claims
Expressions and One-Variable Equations
Variables, equivalence, and balance
Linear Equations and Graphs
Tables, slopes, equations and systems