Teacher quick start
Target
Generate or reject a candidate equivalent ratio by explaining how both labeled quantities change together.
Prerequisites and materials
Count and name the two quantities in a ratio, preserve their order, and use equal groups or multiplication/division with whole numbers. The student lesson includes a short introduction to 'for every' and colon notation. Ordinary paper and pencil. Counters are optional; every group can be sketched.
What to print
Full lesson: first give the separate Readiness page alone; then give Teach the Lesson pages 1–6. Do not distribute the lesson or glossary before the Readiness task. Short check: Check-only pages 1–3, with no model or answers. Follow-up: choose page 1 (Route A) or page 2 (Route B) of Follow-up. Later revisit: pages 1–2 of Later Revisit. The repeated independent tasks in the lesson and Check-only file are the same tasks; use one copy.
Teaching sequence
1. Give the Readiness task before showing the introduction or model; clarify task words without giving the quantities. Use the response to decide whether to teach the ratio-language introduction. 2. Model the worked example with complete groups and the table. 3. Work through Guided tasks 1–3 while gradually removing prompts. 4. Give Independent tasks 1–3 for students to try on their own. 5. Choose Follow-up task 1 or 2 from the student’s response, or clarify a response whose meaning is unclear. 6. Use Later tasks 1–2 after another learning opportunity. The Extension task is optional and explores the whole-group condition.
What to notice and next decision
Record quantity order, shared factor or groups, arithmetic, units, and explanation separately. Same-addend reasoning suggests Route A after clarifying labels. Correct quantities without a reason suggest Route B. An arithmetic slip needs arithmetic support; a blank needs clarification first.
Planning times
Preparation 4–6 minutes; full lesson 35–45 minutes; focused follow-up 8–12 minutes; independent check 4–6 minutes. Later revisit: After another learning opportunity. Planning estimates only; no classroom timing has been measured.
Limits
The three short independent responses show understanding of these tasks and help you choose a next teaching step. Later tasks use new contexts and may have different demands; Later task 2 adds scaling down. An immediate correct answer does not establish lasting understanding or show that a teaching move caused it. Adjust temporary groups as student responses change.
Solutions · introduce and begin
Readiness task
Task: A sticker sheet has 6 stars and 10 circles. Write stars:circles and explain what each number counts. Can you describe the same relationship with two equal groups?
Answer: 6:10. Six counts stars and ten counts circles. The sheet can be seen as two groups of 3 stars and 5 circles, so 3:5 describes the same relationship.
Also accept: A correctly labeled 6:10 description is sufficient readiness evidence even without reducing it to 3:5.
Look for: If the order reverses or 'for every' is unclear, teach the prerequisite introduction before treating scaling as the difficulty.
Worked example
Task: Watch the complete-group model for 3 stars for every 5 circles. Explain why 6:10 keeps the relationship and 6:8 does not.
Answer: 6:10 is two complete 3:5 groups. 6:8 cannot be made from two such groups; 6 stars would need 10 circles.
Also accept: A correctly labeled ratio table, drawing, or multiplication explanation is valid.
Look for: The student-facing model may show its solution; it is never used as an independent check.
Guided task 1
Task: A pattern has 3 stars for every 5 circles. If it has 9 stars, how many circles keep the relationship? Show groups or a table.
Answer: 15 circles. Three groups of 3 stars require three groups of 5 circles.
Also accept: 9:15 or a labeled drawing is valid when the quantities are clear.
Look for: Ask which factor changed the stars and whether it also changed the circles.
Solutions · guided work and first check
Guided task 2
Task: The same pattern has 20 circles. How many stars keep the relationship? Explain starting with the circles.
Answer: 12 stars. Twenty circles is four groups of five, so four groups of three stars give twelve.
Also accept: A table or 20 ÷ 5 = 4, then 3 × 4 = 12 is valid.
Look for: The known quantity is second; do not let a memorized first-column procedure hide its meaning.
Guided task 3
Task: Which pairs keep 3 stars for every 5 circles: 6:10, 6:8, 9:15, 12:20? Explain one pair that works and why 6:8 does not.
Answer: 6:10, 9:15, and 12:20 work; the complete-group factors are 2, 3, and 4. For 6 stars, 6:8 is two circles short of 6:10.
Also accept: Equivalent fraction or ratio-table reasoning is valid if labels and both quantities are handled consistently.
Look for: Distinguish adding a complete 3:5 group from adding the same number to both original counts.
Independent task 1
Task: A flag display repeats a group of 2 triangle flags and 7 rectangular flags. If it has 8 triangle flags, how many rectangular flags should it have? Show why.
Answer: 28 rectangular flags. Eight triangles make four groups of two; four groups of seven rectangles make twenty-eight.
Also accept: A labeled table or four sketched complete groups is valid.
Look for: Look for matching scale in both quantities and an explanation, not only 28.
Solutions · independent check
Independent task 2
Task: A different banner uses 3 square flags for every 8 round flags. Two proposals are 6 squares with 16 rounds and 6 squares with 11 rounds. Which keeps the relationship? Explain why the other does not.
Answer: 6:16 keeps the relationship because both original counts double. 6:11 adds three to each original count; six squares need sixteen rounds, so eleven is five short.
Also accept: Any clear complete-group or shared-factor comparison that rejects 6:11 and explains the quantity mismatch is valid.
Look for: This is a new ratio; it does not supply a useful intermediate pair for Independent task 1. A correct choice without a reason gives limited evidence of the target.
Independent task 3
Task: A strip uses 5 centimetres of striped ribbon for every 8 centimetres of plain ribbon. A version has 15 centimetres of striped ribbon. How much plain ribbon does it need? Explain and keep the units.
Answer: 24 centimetres of plain ribbon. Fifteen is three groups of five centimetres striped, so three groups of eight centimetres plain are needed.
Also accept: A ratio table or scaling both lengths by three is valid.
Look for: This transfers to lengths; a correct number with reversed labels does not fully interpret the relationship.
Solutions · follow-up and first revisit
Follow-up task 1
Task: A model uses 2 triangle counters for every 3 circle counters. Sketch three complete groups. Count both totals. Does 6 triangles with 7 circles meet the rule? Explain.
Answer: Three groups have 6 triangles and 9 circles. The 6-and-7 proposal is short of two circles.
Also accept: A labeled sketch, table, or repeated addition of complete 2:3 groups is valid. Physical counters are optional.
Look for: Use when a learner applies a same-addend rule; verify the quantities were understood first.
Follow-up task 2
Task: A pattern uses 4 square tiles for every 7 round tiles. A larger pattern has 21 round tiles. Find the square tiles. Show the factor in both quantities and explain it in words.
Answer: 12 square tiles. Twenty-one round tiles are three groups of seven, so three groups of four square tiles give twelve.
Also accept: A correctly labeled table, sketch, or division-then-multiplication argument is valid.
Look for: Use when quantities are correct but the explanation is unclear; do not replace the explanation with a filled sentence alone.
Later task 1
Task: A print display uses 4 small prints for every 7 large prints. With 28 large prints, how many small prints keep the relationship? Explain starting with the large prints.
Answer: 16 small prints. Twenty-eight large prints are four groups of seven; four groups of four small prints give sixteen.
Also accept: A labeled table or factor-of-four explanation is valid.
Look for: This later task starts with the second quantity and uses a new context; it may have different demands from earlier checks.
Solutions · later revisit and extension
Later task 2
Task: A craft plan uses 6 metres of cord for every 9 metres of ribbon. A smaller version uses 2 metres of cord. How much ribbon keeps the relationship? Explain whether quantities must become larger to remain equivalent.
Answer: 3 metres of ribbon. Dividing both quantities by three keeps the relationship; equivalent ratios may use smaller quantities.
Also accept: A correctly labeled 2:3 comparison or one-third scaling argument is valid.
Look for: This later task adds scaling down, so it may have different demands from earlier checks.
Extension task
Task: A pattern is built only from complete groups of 3 stars and 5 circles. Can it contain exactly 30 stickers with none left over? What about 32? Explain the whole-group rule.
Answer: Each complete group has eight stickers. Thirty cannot be made from a whole number of these groups. Thirty-two is four groups: twelve stars and twenty circles.
Also accept: A count by eights or a whole-number division argument is valid; fractional stickers do not meet the complete-group rule.
Look for: This is a boundary question for discrete complete groups, not another repetitive scaling item.
Choose one follow-up
Build complete matched groups · Follow-up task 1
Use when: A response uses the same addend on both unequal quantities, after the learner has correctly named the quantities.
Say and do: One complete group needs both parts. Sketch the first 2-triangle-and-3-circle group, then a second and third. Count each type. Which part is missing from the proposed 6-and-7 model?
What to look for: Look for a matching number of complete groups and an explanation that both parts change together.
Explain the shared factor · Follow-up task 2
Use when: The numerical quantities are correct, but the response does not reveal why they preserve the relationship.
Say and do: Start with the quantity we know. How many groups of seven are in twenty-one? What must those same groups mean for the square tiles? Show this in a table, then explain it in your own words.
What to look for: Look for the factor connected to both labeled quantities, not a memorized number pair.
Response: 4:9 works because I added two
Clarify: Ask what each number counts. Next move: Use Follow-up task 1 to rebuild complete matched groups if the quantities are understood.
Response: Correct quantities without a clear reason
Clarify: Ask the learner to show one group or a common factor. Next move: Use Follow-up task 2 to connect a table to the explanation.
Response: Correct setup with an arithmetic slip
Clarify: Check the calculation or estimate. Next move: Support computation while preserving valid ratio reasoning.
Response: Reversed order or unclear units
Clarify: Ask which quantity is first and what each number counts. Next move: Return to the ratio-language introduction before reteaching scaling.
Response: Blank or uninterpretable
Clarify: Check access, directions, response mode, and time. Next move: Clarify first; do not infer a misconception from a blank.
Supports and teaching scope
Understanding student responses
For each independent task, record separately: labeled quantity order, a valid shared group/factor, arithmetic, units when present, and a reason. Do not collapse three responses into a broad ratio-ability score or diagnosis. A solution different from the model is acceptable if its relationship is correct and explained.
Support without hiding the target
Barrier: Holding both quantities in working memory
Support: Keep labels visible and sketch one complete group before making a table. Constant: Both parts must scale together and be explained. What this shows: No change to the ratio target when the sketch is learner-created.
Barrier: Transcribing a written explanation
Support: Permit an oral or dictated explanation with a labeled sketch; record response mode. Constant: The explanation must connect both quantities. What this shows: Does not show independent written mathematical communication.
Barrier: Arithmetic load obscures reasoning
Support: Allow a calculator after the group/factor is identified and record its use. Constant: The learner still chooses the relationship and labels quantities. What this shows: The result cannot be used to infer unaided arithmetic accuracy.
Standards boundary
6.RP.A.3a: Some tasks address equivalent-ratio tables and missing whole-number values; this lesson does not assess coordinate plotting or the full standard.
Classroom use
These original tasks use fictional contexts. Groups and tables are editable in the Word files.