Contents
- Start with the decision, not the task format
- Compare the two purposes
- Plan a shared equal-groups learning goal
- Copy-ready in-lesson formative check
- Interpret responses and decide what to teach next
- Copy-ready end-of-sequence assessment
- Record a bounded end-of-sequence judgement
- Reusable assessment planning template
- Related resources
Start with the decision, not the task format
Plan two uses of evidence for the same learning goal: one to decide what to teach next, and one to judge what pupils can demonstrate at the end of a teaching sequence. The equal-groups sequence below includes pupil task sheets, expected answers, response-based teaching decisions and a reusable planning template.
The key distinction is purpose. The NSW Department of Education defines formative and summative assessment by how the information collected is intended to be used. Formative use directly informs next steps in learning; summative use evaluates achievement against learning objectives or curriculum standards after an extended period of teaching. Summative evidence can also be used formatively to guide further learning. Purpose of assessment — NSW Department of Education
Do not classify an assessment simply because it is a quiz, a conversation, marked work or an end-of-lesson task. Ask instead: “What decision will I make from these responses?” Timing provides context, but neither timing, format nor the presence of a grade is sufficient to define the purpose.
Compare the two purposes
Formative use
Question: What does this pupil currently understand, and what should I ask or teach next?
Planned action: Inspect the model and explanation, ask a clarifying question, then adjust teaching before checking again.
Summative use
Question: What has this pupil demonstrated against the stated learning goal at the end of the sequence?
Planned action: Compare the evidence with the goal and record a judgement, including any support used.
Combined use
Record the end-of-sequence judgement first, then use the identified gap to plan subsequent teaching. Keep those two decisions explicit.
The classroom examples and pupil responses below are hypothetical. For example, a pupil’s equal-groups drawing could prompt an immediate request to distinguish the number of groups from the number in each group. That is a formative use. A comparable drawing collected at the end of the sequence could contribute to a judgement about the learning goal. That is a summative use. Calling both activities a “worksheet” does not tell you which decision is being made.
The Australian Curriculum describes formative assessment as deliberately focused on specific content and skills, using evidence of what pupils currently know and can do to guide teaching. Its assessment guidance also connects summative assessment with expected learning described in achievement standards. Aligning curriculum and assessment — Australian Curriculum
Plan a shared equal-groups learning goal
Use this learning goal for both checks: Represent and explain equal groups by identifying the number of groups and the number in each group, finding the total, and connecting the model to repeated addition and multiplication.
NCETM’s primary equal-groups materials distinguish equal from unequal groups, require both the number of groups and the size of each group to be defined, and connect equal-group representations with repeated addition and multiplication expressions. Structures: multiplication representing equal groups — NCETM
Prepare counters, small mats or paper circles, and drawing materials. Begin by modelling an equal-group arrangement. Point to a whole group while naming the number of groups, then point within one group while naming its size. Connect each group to one term in a repeated-addition expression and explain how a multiplication expression describes the arrangement.
Look for four connected pieces of evidence: the model matches the situation; the pupil distinguishes group number from group size; the total is correct; and the addition and multiplication expressions are explained in relation to the groups. Use an unequal arrangement as a contrast, asking whether one repeated group size describes all the groups.
Give the formative task during teaching. Use the responses to choose follow-up teaching, then offer further practice with different arrangements. At the end of the sequence, use the summative task: it keeps the mathematical demand but changes the quantities and context.
Copy-ready in-lesson formative check
Name: ____________________ Date: ____________________
Equal groups: show and explain
You may use counters, draw, or explain aloud to your teacher.
1. There are 3 trays. Each tray holds 4 counters.
Build or draw this arrangement.
2. Complete:
There are ______ groups.
There are ______ counters in each group.
There are ______ counters altogether.
3. Write an addition expression that matches the groups.
________________________________________________
4. Write a multiplication expression for the total.
Explain what each number tells us about your arrangement.
________________________________________________
________________________________________________
5. A different arrangement has 3 trays holding 4, 4 and 2 counters.
Are these equal groups? Explain how you know.
________________________________________________
Teacher answer guide
1–2. The model should show 3 distinct groups with 4 counters in each; the total is 12.
3. 4 + 4 + 4 = 12 matches the three groups, with one addend for each tray.
4. 3 × 4 = 12 or 4 × 3 = 12 is numerically correct. Ask the pupil to identify the 3 trays and the 4 counters per tray in their explanation rather than relying on factor order alone.
5. The groups are unequal because one tray has 2 counters while the others have 4. Although 4 + 4 + 2 = 10 gives the total, it does not repeat one equal group size across all three trays.
England’s primary mathematics programme of study includes teaching that multiplication of two numbers can be done in either order. A reversed multiplication expression is therefore not, by itself, evidence that a pupil has confused the quantities in an equal-groups situation. National curriculum in England: mathematics programmes of study — Department for Education
During this check, note the pupil’s first response before giving a mathematical hint. Record any explanation given orally and any prompt that changes the answer. Keep this evidence for deciding the next teaching move, rather than turning the check into a pass mark.
Interpret responses and decide what to teach next
Response: The pupil builds 3 groups of 4, counts every counter and says “12”.
Ask next: “Show me one group. How many groups are there? Where can you see each 4 in your addition?”
Teaching decision: Do not treat counting all as a misconception. If the pupil identifies the structure accurately, connect that valid counting strategy to 4 + 4 + 4 and a multiplication expression. If the explanation remains unclear, keep the model visible while exploring it.
Response: The pupil writes 3 + 4 = 7.
Ask next: “What does the 3 describe? What does the 4 describe? Can you show all the counters on the trays?”
Teaching decision: If the pupil is adding the number of trays to the number per tray, return to the model. Touch each tray in turn and record its 4 counters as one addend. Then ask the pupil to match a fresh arrangement to an addition expression.
Response: The pupil draws 4 groups of 3 and writes 12.
Ask next: “Your total is 12. Where are the 3 trays in your drawing, and how many counters did you put on each?”
Teaching decision: Separate a correct total from a model that matches the situation. Rebuild the stated arrangement and compare the two models, naming what changed and what stayed the same.
Response: The pupil draws the correct model and writes 4 + 4 + 4 = 11.
Ask next: “Show me how you found the total.”
Teaching decision: Check the calculation without assuming the grouping concept is missing. Ask the pupil to recount or combine the groups, then give another short total-finding opportunity while retaining the same structure.
Response: The pupil calls the 4, 4 and 2 arrangement equal because “there are three trays”.
Ask next: “Compare what is inside each tray. What must be the same for these to be equal groups?”
Teaching decision: Contrast equal group size with equal-looking containers. Have the pupil change the counters to make equal groups and explain the change.
Response: The pupil accurately models, calculates and explains both arrangements.
Ask next: “Make a different equal-group arrangement with a total of 12. What changed?”
Teaching decision: Ask for a new explanation rather than more copies of the same calculation. Later, check the shared goal with the end-of-sequence task.
After the follow-up teaching, use a brief fresh check: “Make 2 groups of 5. Tell me the group number, group size and total, then write the matching addition.” Note whether the pupil now makes the connection without the earlier prompt.
Copy-ready end-of-sequence assessment
Name: ____________________ Date: ____________________
Equal groups: end-of-sequence task
You may use counters or drawings. Explain your thinking in writing or aloud.
1. There are 4 pots. Each pot holds 5 buttons.
Build or draw this arrangement.
2. Complete:
There are ______ groups.
There are ______ buttons in each group.
There are ______ buttons altogether.
3. Write an addition expression that matches the groups.
________________________________________________
4. Write a multiplication expression for the total.
Explain what each number tells us about your arrangement.
________________________________________________
________________________________________________
5. Another arrangement has 4 pots holding 5, 5, 5 and 3 buttons.
Are these equal groups? Explain how you know.
________________________________________________
Teacher answer guide
1–2. The model should show 4 distinct groups with 5 buttons in each; the total is 20.
3. 5 + 5 + 5 + 5 = 20 matches the four pots.
4. 4 × 5 = 20 or 5 × 4 = 20 is numerically correct. The explanation should identify 4 as the number of pots and 5 as the number of buttons in each pot.
5. These are not equal groups: three pots have 5 buttons each and one has 3. The total is 18, but a total alone does not establish equal group size.
Before using the task, decide what pupils may access and record those conditions. For this goal, allow counters or drawings and accept an oral explanation. Read the wording aloud if needed without explaining the mathematics. Collect the initial response before offering teaching prompts; note any mathematical support given.
Record a bounded end-of-sequence judgement
Record the evidence by component, not just by total correct. For each pupil, note:
Model: Does it match the stated groups?
Group description: Are group number and group size distinguished?
Total: Is it correct?
Expressions: Do repeated addition and multiplication connect to the model?
Explanation: Does the pupil identify why the contrasting arrangement is unequal?
Conditions: What materials, oral responses or prompts were used?
Judgement: What does this evidence show about the stated goal?
Next teaching: What remains to be addressed?
Possible record: “Demonstrated the stated goal using counters, without mathematical prompts. Built 4 groups of 5, named both quantities, found 20, linked the model to repeated addition and multiplication, and explained why the comparison arrangement was unequal.”
Possible record: “Demonstrated equal-group structure and both representations. Wrote 5 + 5 + 5 + 5 = 19, so total-finding accuracy remains unresolved. Recheck calculation separately.” This preserves the evidence of understanding rather than treating the whole response as simply wrong.
Possible record: “Correct total of 20, but drew 5 groups of 4 and could not locate the 4 pots in the model. Matching group number and group size to the situation is not yet demonstrated independently. Revisit model-to-language connections.”
These tasks and recording criteria support a classroom judgement about this specific learning goal. They are not standardised assessments, numerical mastery thresholds or a basis for assigning a whole-mathematics attainment label. Use your applicable curriculum expectations when making broader judgements.
Reusable assessment planning template
Learning goal:
________________________________________________
Relevant curriculum expectation:
________________________________________________
Evidence sought: What must pupils show, say, make or write?
________________________________________________
Formative purpose: What teaching decision will this evidence inform?
________________________________________________
In-lesson prompt and expected response:
________________________________________________
________________________________________________
Possible response → clarifying question → teaching action:
________________________________________________
________________________________________________
Fresh check after teaching:
________________________________________________
Summative purpose: What judgement will I make at the end?
________________________________________________
Matched but non-identical end-of-sequence prompt:
________________________________________________
________________________________________________
Expected answer and acceptable representations:
________________________________________________
Assessment conditions and support to record:
________________________________________________
Evidence by component:
________________________________________________
End-of-sequence judgement against the goal:
________________________________________________
Next teaching or further evidence needed:
________________________________________________
Before teaching, check the link across the plan: the goal determines the evidence; the assessment purpose determines how you will use it; and the response determines either your next teaching move or your recorded judgement. If you cannot name that final decision, revise the plan before adding more questions.
Related resources
Common formative assessments: a primary mathematics planning kit
Formative assessment for science: three materials question cards
