Proportional Relationships: From Ratios to Models

Use a complete sequence to move from equivalent ratios and per-one rates to recognizing and explaining proportional relationships. The new collection supplies first teaching, response lessons, different independent checks and changed-setting transfer with separate complete keys.

Choose the entry point

1. Paired quantities need teaching: use Equivalent Ratios: Scale Both Quantities. Keep its existing later checks for their intended use.
2. Per-one direction needs teaching: use Find and Interpret Per One. Selected unchanged first-teaching files are included in this ZIP. Its original check is a prerequisite check, rather than evidence of the new relationship-testing target.
3. Unit rates are secure: begin the new Test and Explain Proportional Relationships sequence. Use Starting Task, the three main teaching lessons and Independent Check, then select the actual response teaching and later checks.

Teach the relationship rather than one calculation

A learner can find a unit rate correctly from one pair and still overstate what it proves. The supplied teaching compares every relevant positive-input pair, handles the zero pair separately, and connects those judgments to the rule, graph and quantities. A straight line with a starting amount can have equal increases without proportional totals. A curve may pass through the origin and still fail the common-multiplier test.

The model lessons provide actual tables, contrasting examples, teacher language, guided tasks and individual practice. They introduce coordinates and the meaning of the origin and one-unit point. Whole-number piece counts remain discrete; a time model can permit a continuous line segment. No division by zero is needed.

Choose teaching from the explanation

• Only one pair is tested: use Test Every Displayed Pair to challenge a candidate rule with a different row.
• Equal changes settle the claim: use Check the Starting Value to distinguish total output from output added.
• Units or coordinates are unclear: use Name the Quantities and Points and return to per-one foundations if needed.
• Representations agree but meaning is thin: use Connect Your Representations to explain the constant, one-unit point and domain.
• A small record becomes an unrestricted prediction: use What a Model Establishes and One Pair Two Possible Rules to state what further condition is needed.

The teacher guide includes the models, task-specific answers, exact fractions or decimals, defensible alternatives and questions that investigate uncertainty before selecting teaching. Do not infer a lasting misconception from one missing response.

Keep checks and transfer useful

Use Starting Task before showing its discussion or key. Teach the three main lessons, then give Independent Check alone. After a selected response lesson and another learning opportunity, use the different Follow-Up Check. Save Later Transfer for a later occasion: it changes to ribbon mass with a bag and square side versus area. The checks use different numerical data; they have not been calibrated to equal difficulty.

Record previous exposure, help, calculator or model use, response method and any target change on the local decision record. A shown or supported task is practice. A finite set of matching ratios permits a conditional prediction; it does not prove all unseen readings or behavior beyond the stated domain.

Teacher Start Here and complete package

Printable orientation · Editable Word · HTML text. The teacher guide gives all task-specific answers and local interpretation guidance.

Open all fifteen document sets. The earlier unit-rate collection remains useful for comparing stated rates and whole-pack purchases. A least feasible whole-pack cost answers a different question from proportionality of an input-output relationship.

Preparation and classroom use

Read Start Here and the teacher guide, select the student pages, and use ordinary paper and pencils. Read-aloud, enlarged text, extra processing time or an agreed oral, drawn or labeled-table explanation can preserve the reasoning target. Every graph has numerical descriptions and labeled tables. Quarter values can be placed between integer grid lines; exact fractions or decimals are accepted. Supplying a classification, ratio, rule or point changes the evidence collected.

This is a selected Grade 7 supplement after labeled equivalent-ratio and per-one reasoning. It references recognition and representation in CCSS 7.RP.A.2; it does not cover the full standard, grade or course. All numerical situations and example responses are original constructed teaching material. Classroom effectiveness has not been established.

No account, payment or student upload is needed. Print or adapt the original material under the Terms of Use; keep attribution, identify local changes and check Word print preview after editing. Report a resource issue.

Find the rate when it is not supplied

When quantity, coordinate and proportional reasoning are secure, continue to the selected Grade 8 Linear Relationships collection. It teaches change per input unit from unequal intervals, initial-value recovery, signed and zero cases, representations and rate versus output comparisons. Use its new entry task and record the added arithmetic demands; keep this collection’s later checks and exposure records for their original target.