Modeling Multi-Step Situations

Overview

In lesson 22 of 26, students apply linear patterns and algebraic expressions to solve multi-step contextual problems. Building on prior work with tables, variables, expressions, and graphs, they choose a useful representation, explain their reasoning, and interpret the answer in context.

Learning intentions

Students will:

• model a multi-step situation using a table, diagram, expression, or graph;
• identify the variable, operations, and order of steps in a problem;
• solve and check a linear expression in a familiar context;
• communicate an answer using calculations, evidence, and a complete sentence.

Success criteria

• I can identify what the variable represents and what each number means.
• I can choose and create a helpful model.
• I can calculate accurately and check whether my answer is reasonable.
• I can explain what my answer means in the situation.

Curriculum links

• Patterns and Relations — oral and written patterns and their corresponding relations.
• Patterns and Relations — constructing tables, graphing relations, and analysing graphs to solve problems.
• Patterns and Relations — evaluating expressions when variables have given values.
• Mathematical processes — communication, connections, mental mathematics and estimation, problem solving, reasoning, technology, and visualization.

Lesson structure (50 minutes)

1. 0–5 min · Attention hook and retrieval. Display a visual of a colony store receipt with a fixed fee and repeated item cost using the hook and retrieval slides. Ask, “What changes, what stays the same, and how could we represent the total?” Students silently identify the variable, then share one possible table, diagram, or expression. Quickly review terms: variable, constant, coefficient, expression, and relation.

2. 5–12 min · Explicit modelling. Use the worked-example slides to model: “A community supper charges a $6 booking fee and $4 for each meal. A family buys 5 meals and adds a $3 donation. What is the total?” Think aloud: underline facts, name the variable, sequence the operations, and write (6 + 4m + 3). Substitute (m=5), calculate $29, and interpret the answer. Students complete matching guided notes on the guided notes and problem-solving mat and hold up a chosen representation.

3. 12–20 min · Concrete guided practice. Give pairs the Algebra Balance Mat to support step-by-step organisation using the algebra balance mat. Work through a second example with counters or drawn groups: “A student has $20, spends $5, then earns $3 for each of 4 jobs.” Students build or sketch the situation, write (20-5+3j), substitute (j=4), and explain the result. Pause for checks after each step; students use the sentence frame, “First…, then…, so the expression is…”.

4. 20–30 min · Contextual task. Distribute the Mathology contextual task through the Mathology-style contextual task section. Present one accessible problem at a time, reading it aloud while students mark known information and circle the question. In pairs, students complete the problem-solving mat: understand, plan a representation, solve, check, and interpret. Offer a choice of table, diagram, expression, or graph. Confer with each student and prompt rather than supply the next step.

5. 30–35 min · Video connection and discussion. Show a short, pre-selected age-appropriate segment from the teacher’s algebraic-expressions word-problems search using the video viewing and discussion slide. Students record one strategy they notice and compare it with the class method. Stop before the solution when possible and ask, “What should the variable mean?” and “Which operation comes first?”

6. 35–44 min · Targeted and independent practice. Students complete assigned Mathletics review questions, beginning with substitution and one-step expressions before attempting multi-step problems. Early finishers complete the independent questions on the independent practice section. The teacher runs a short conference group for students needing reach-back instruction, using a familiar fee-plus-items context and a pre-filled table.

7. 44–50 min · Evidence checkpoint and exit reflection. Collect one completed problem-solving mat from each student and assess the selected task for model, calculations, explanation, and interpretation. Students complete the final reflection on the exit reflection: “My variable represents…”, “My model helped because…”, and “My answer means…”. Invite each student to state a confidence rating and one next step.

Resources

• the complete modelling and discussion slide deck
• the guided notes, problem-solving mat, contextual task, practice, and reflection
• the algebra balance mat
• Mathology contextual task materials
• Mathletics devices and assigned review set
• Board or chart paper and markers
• Counters, linking cubes, or coins
• Projector, speakers, and a pre-selected video segment
• Calculators for students who require them

Assessment

• Knowledge and Understanding: During retrieval and modelling, check whether students identify variables, constants, coefficients, operations, and the meaning of an expression. Record progress when a student correctly identifies and evaluates an expression.
• Mental Math and Estimation: Ask students to estimate totals before calculating and explain whether an answer is reasonable. Record progress when a student uses an efficient estimate or checks the size of an answer.
• Problem Solving: Use the collected mat as the primary evidence. Assess whether the student selects an appropriate model, represents the situation accurately, calculates correctly, explains the strategy, and interprets the answer in context. Note the level of independence and prompting required.

Differentiation

• Provide chunked text, bolded key facts, oral reading, repeated directions, dyslexia-friendly sans-serif print, generous spacing, and the sentence frames “The variable represents…” and “Therefore…”.
• Use familiar colony, store, supper, travel, and work contexts; reduce the number of steps or use smaller data sets for modified-program students while retaining the same modelling process.
• Pre-fill the first row of a table, provide a partially completed diagram, colour-code fixed and changing quantities, and allow counters, drawing, calculator use, or speech-to-text.
• Seat distractible students near the teaching space, give them a defined role such as reader or checker, use short timed work intervals, and provide frequent private check-ins. Challenge confident students to represent the same situation in two different ways and explain which is more efficient.