Grade 2 · Mathematical Practice · MAT-02-MATHPRACTICE-001
Represent a Problem with Objects, Drawings, Equations, and Words
Represent the same problem with concrete objects, drawings, equations, and words.
Learning objectives
What you should be able to do
- Represent the same problem with concrete objects, drawings, equations, and words.
- Explain how each representation encodes the same quantities and relationships.
- Use multiple representations to solve or verify a problem.
Prerequisite check
Make sure the foundation is ready.
Checks Grade1 multiple-representation foundation.
Learn
Build the idea from meaning, not memorization.
Representations are different views of the same mathematics
Objects make quantities tangible, drawings preserve the structure visually, equations compress relationships into symbols, and words explain meaning. A strong solution can move between these forms.
The representations must agree
If a drawing shows 12 objects but the equation totals 14, something is inconsistent. Agreement among representations is a built-in error check.
Choose representations that expose the structure
Base-ten blocks are useful for regrouping, bar models for part-whole problems, number lines for distance/change, and fraction models for equal shares.
Words should explain relationships
A complete explanation states what each number represents and why the operation/model fits, rather than simply restating the answer.
Interactive math lab
Change it. See what stays true.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- objects to equation
- drawing to words
- equation to model
- find mismatch
Worked examples
Twenty different ways to see the concept work.
23+18 with blocks.
2T3O+1T8O ->4T1O after regroupObjects show place value.
23+18 drawing.
tens/ones sketchSame regrouping visually.
23+18 equation.
23+18=41Symbolic form.
23+18 words.
23 items plus18 more makes41Meaning.
52−27 blocks.
5T2O ->4T12O ->25Trade visible.
52−27 number line.
52 back27 to25Distance/change view.
Part-whole: 35 total,12 red.
bar with whole35,part12,missing23Structure visible.
Equation for same.
35−12=23Missing part.
Graph compare8 vs5.
bar values +8−5=3Data to equation.
Length12cm+8cm.
segments totaling20cmMeasurement model.
Money65¢ paid100¢.
coins/number line;100−65=35Change.
Half model.
shape split2 equalFraction relation.
Array3 rows4.
drawing;4+4+4=12Equal groups.
Time3:25.
clock drawing+digital notationSame time.
Unknown□+9=17.
balance/model + equationMissing addend.
Why blocks better than random dots for368+145?
Place values visibleEfficient representation.
Why words still needed?
Explain what symbols meanCommunication.
Mismatch drawing7 but equation3+5=8.
errorRepresentations disagree.
Can two drawings both be valid?
YesDifferent models may preserve same structure.
Why switch representations?
One form can reveal what another hidesSupports reasoning/checking.
Practice
Guided practice with hints
01Show34+27 three ways.
Hint: Use blocks,drawing,equation.
Answer: Any consistent representations totaling61
02Show60−18 with model.
Hint: Regroup tens.
Answer: 42
03Represent compare9 vs6.
Hint: Bars or objects.
Answer: difference3
04Represent one fourth.
Hint: Equal4-part model.
Answer: one shaded part
05What if equation and drawing disagree?
Hint: Recheck both.
Answer: Correct inconsistency
06Why name number meanings?
Hint: Connect symbols to context.
Answer: Explanation
Practice
Independent practice
01Representation for45+16
Answer: Any valid objects/drawing/equation/words matching61.
02Representation for70−25
Answer: Any valid model showing45.
03Graph compare7 vs4 equation
Answer: 7−4=3
04Array2x5 repeated addition
Answer: 5+5=10
05One third model
Answer: 1 of3 equal shares
06Why multiple forms
Answer: To communicate and check reasoning.
07Agreement requirement
Answer: All forms must encode same quantities.
08Best model for regrouping
Answer: Base-ten/place-value model.
Common mistakes
Learn to catch the error, not just the answer.
Every drawn element should represent a quantity, unit, or relationship.
Trace each symbol back to represented quantities.
Explain what quantities mean and why operation fits.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Explain homework solutions to another learner.
- Use drawings/models to check real-world measurement, money, or data problems.
Challenge problems
Can one problem have multiple valid equations?
Yes, related equations can describe same relationships.
Why might number line and blocks lead to same answer differently?
They encode different mathematical structures while preserving quantity.
Create representation mismatch deliberately, then diagnose.
Any example where one form encodes different quantity.
Which representation best shows52−27 regrouping and why?
Base-ten blocks because trading1 ten for10 ones is visible.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 6
Game mode
Representation Relay
Translate the same problem across objects, drawings, equations, and words.
Practice
Mastery check
01Name4 representation types
Answer: Objects,drawings,equations,words
02Why must agree
Answer: They describe same math.
03Best regrouping model
Answer: Base-ten blocks
04Best equal-share model
Answer: Fraction partition
05If drawing/equation disagree
Answer: Recheck/correct
06Why words matter
Answer: Explain number meanings and reasoning.
07Can two valid models differ
Answer: Yes
08What makes representation useful
Answer: It preserves relevant quantities/relationships.
Terminology
Words to know
- representation
- A way to show a mathematical idea or relationship.
- model
- A representation used to reason about a situation.
- symbolic
- Expressed with numbers and mathematical symbols.
- consistent
- Agreeing in value and meaning.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Standards for Mathematical Practice MP1,MP2,MP4,MP5
Model, represent, and reason quantitatively.