Math Routines and Representations

Overview

In this first lesson of the 30-lesson unit Essential Math Problem Solving, students establish a repeatable routine for understanding word problems before calculating. They connect prior work with rational numbers, equations, and inequalities to selecting a useful representation such as a drawing, number line, table, or labeled model.

Learning intentions

Students will be able to:

• Identify what a mathematical problem is asking.
• Distinguish important information from unnecessary information.
• Choose a helpful representation for a real-world problem.
• Use estimation or reasoning to check whether an answer is reasonable.

Success criteria

• I can underline the question the problem is asking.
• I can circle important numbers and units.
• I can draw or write a model that represents the situation.
• I can explain how my representation helps me solve or understand the problem.

Curriculum links

• Expressions and Equations — solve multi-step real-life problems using positive and negative rational numbers, tools, and estimation.
• Expressions and Equations — use variables to represent quantities and construct equations or inequalities from situations.
• Expressions and Equations — interpret word problems and connect representations to mathematical reasoning.

Lesson structure (30 minutes)

1. 0–3 min · Bell work routine. Display the Day 1 review problem from the introduction and bell-work slides and distribute the four-day problem-solving cycle sheet. Students work independently for no more than three minutes, underlining the question and circling important numbers before attempting a solution.

Bell work problem: “A hiking trail is 6.5 miles long. Jordan has walked 2.75 miles. How many miles remain?” Students may use a drawing, number line, or equation. Do not correct every step yet; quickly notice which representation students choose.

2. 3–7 min · Notice and discuss. Revisit the bell work on the bell-work discussion slide and ask, “What did the problem ask us to find?” and “Which information helped us answer it?” Students compare their markings with a partner and explain one choice using the sentence stem, “I circled ___ because ___.”

3. 7–12 min · Teach the routine. Model the three-part routine on the routine modeling slides using a short problem: “A game costs $18.50 and is on sale for $5 off. What is the sale price?” Think aloud: underline the question, circle $18.50 and $5, then draw a bar model or write an equation such as (18.50 - 5 = p). Students copy the markings and identify the known quantities and unknown.

4. 12–20 min · Guided representation practice. Read the following problem aloud and show it on the guided-practice slide: “A school club has a $40 starting balance and spends $6.50 on each of four posters. How much money remains?” Students mark the problem on their worksheet, then choose one representation: labeled drawing, number line, table, or equation. Confer with pairs and ask, “What does each number represent?” and “How does your model show the operation?”

5. 20–26 min · Partner comparison and reasoning. Display the representation choices on the partner-discussion slide. Pairs compare their models with another pair, checking whether both show the same quantities and relationship. Students explain which representation is most helpful and estimate first: approximately (40 - 4(7) = 12), so an answer near $14 should be reasonable.

6. 26–30 min · Exit check and reflection. Use the plenary and exit-ticket slide to display: “A submarine is 12.5 meters below sea level and rises 4.75 meters. What is its new position?” Students independently underline the question, circle important numbers, and draw or write one model on the bottom of the worksheet. Invite two students to share different representations, then collect the sheets to assess the routine.

Resources

• the complete 30-minute routine deck
• the four-day problem-solving cycle sheet
• Pencils and colored pencils or highlighters
• Individual whiteboards or scratch paper
• Optional number lines
• Optional counters or integer tiles
• Large-print copies with increased spacing
• Text-to-speech or screen-reading access

Assessment

• During bell work, check whether students can identify the question and relevant quantities within three minutes.
• During guided practice, listen for accurate explanations of what numbers and units represent; ask students to revise models that do not match the situation.
• Collect the final worksheet response and look for all three routine actions: underlined question, circled important numbers, and a valid drawing or written model.

Differentiation

• Read each problem and direction aloud, present one sentence or chunk at a time, and preview key vocabulary such as “remain,” “balance,” “spends,” and “position.”
• Provide a dyslexia-friendly version with large, clear print, increased spacing, short line lengths, reduced visual clutter, and text-to-speech. Permit students to explain their model orally or use a scribe.
• Offer a choice board limited to a drawing, number line, table, or equation. Provide partially completed number lines, labeled bar-model frames, manipulatives, or color coding for known and unknown quantities.
• For students ready for greater challenge, require a second representation for the guided-practice problem and have them explain why both representations work and which is more efficient.

Extension

• Write a new problem with the same mathematical structure as the guided-practice task, then represent it in two different ways.
• Add a positive or negative rational number to the exit problem and explain how the number line or equation changes.