Part 2- MTP
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MTP 1 — Establish mathematics goals to focus learning
Explanation
Teachers set explicit, clear, and attainable learning goals for lessons and sequences that are connected to curriculum standards and that guide instructional choices.
Goals are communicated to students in student-friendly language and revisited during and after instruction so learning stays focused.
Alignment to Standards of Mathematical Practice
MP1 (Make sense of problems and persevere): Clear goals help students understand what problem-solving skills or mathematical reasoning they are expected to develop.
MP6 (Attend to precision): Goals that include precision expectations (notation, explanation) encourage students to be exact in communication and procedures.
Classroom Implementation Examples
Example 1 — Goal-driven exploration: “Understand and use equivalent fractions”
Objective (teacher/student-friendly): “Today you will explain why fractions like and are equivalent and use models to create equivalent fractions.”
Sequence:
1. Activate prior knowledge: quick warm-up—shade on a fraction strip.
2. Introduce goal and success criteria: “I can show two fractions are equivalent with a model, an equation, and words.”
3. Exploration task: Students use fraction strips or paper-folding to find equivalents for and record results.
4. Reflection: Students compare model, equation (multiply numerator & denominator by same number), and explanation.
Example problem: “Use fraction strips to show two fractions equivalent to . Write an equation and a one-sentence explanation.”
Mock student work:
Model: three-strip diagram showing , and .
Equation: .
Explanation: “Multiplying top and bottom by 2 keeps the fraction the same size, so = .”
Manipulatives/visuals: fraction strips, paper circle cut into fourths and eighths, or a digital fraction-strip tool.
Example 2 — Multi-lesson goal with formative checkpoints: “Develop fluency with multiplication facts through strategies”
Objective: “I can use strategies (doubling, place-value decomposition, and known facts) to multiply within 100 efficiently.”
Sequence:
1. Explicit goal posted for the week, with success criteria (strategy use, correct answer, explanation).
2. Mini-lessons teach specific strategies (e.g., doubling for x4, decomposing factors).
3. Daily 5-minute checks: quick problem sets where students label which strategy they used.
4. Culminating task: strategy portfolio where a student shows the same product solved three ways with explanation.
Example problem: “Compute using two different strategies. Label each strategy and explain why it works.”
Mock student work:
Strategy 1: (decomposition/doubling)
Strategy 2: Known fact extension:
Manipulatives/visuals: arrays on grid paper, linking cubes arranged in rows, strategy anchor chart.
MTP 2 — Implement tasks that promote reasoning and problem solving
Explanation
Use rich tasks that require conceptual reasoning, multiple solution paths, justification, and connections among ideas rather than only procedural practice.
Tasks should be open-ended or have low entry/high ceiling, allowing access and extension.
Alignment to Standards of Mathematical Practice
MP2 (Reason abstractly and quantitatively): Rich tasks require translation between context and symbols, and sense-making.
MP3 (Construct viable arguments and critique reasoning): Tasks that ask for explanations and justifications develop argumentation skills.
Classroom Implementation Examples
Example 1 — Open problem: “Which is larger: or ?”
Purpose: students must reason about sizes of fractions using number lines, area models, or cross-multiplication with explanation.
Task format: small-group debate—each group presents one representation and justification.
Example problem: “Decide which is larger: or . Show two different representations and explain why your method proves the answer.”
Mock student work:
Student A: Converted to common denominator , so is larger.
Student B: Drew number line with equal-length segments and plotted both fractions; labels and work justify the comparison.
Manipulatives/visuals: number-line strips, fraction bars.
Example 2 — Multi-step reasoning: “Design a fence”
Grade-level: upper elementary
Task: “You have 30 meters of fencing; design a rectangular garden with integer side lengths (meters) that maximizes area. Explain your reasoning and prove it's maximal among integer sides.”
Student activity:
1. List factor pairs of perimeter constraint (2L + 2W = 30).
2. Calculate areas and reason about symmetry (L close to W gives larger area).
3. Argue and prove using comparison table or algebra: area = L(15-L); find integer L maximizing product.
Mock student work:
Table with L from 1 to 14, computed W and area, showing max at L = 7, W = 8 (area 56).
Short proof referencing quadratic shape: area as function of L is symmetric about , so integer max at 7 or 8.
Manipulatives/visuals: grid paper to build rectangles; graph of area vs L.