Change in Quantity: Pictorial and Symbolic
Warm-up
Show a ten-frame with 7 counters. Write 7 + ? = 10. How many empty spaces? (3.) So 7 + 3 = 10. Now show a double ten-frame with 14 counters. Write 14 + ? = 20. (6 empty.) This visual-to-symbolic connection is the entire lesson in miniature.
Explore
Change problem cards: each card shows a situation with one quantity unknown. Students must: (1) draw the ten-frame or number line showing the situation, (2) write the equation with a box for the unknown, (3) solve by examining the visual model. Three card types for each structure.
Formalize
Bring together the three structures with a word problem context: a jar had some marbles. I added 15. Now there are 34. How many were there to start? Unknown start: ? + 15 = 34. Draw the number line: end at 34, jump back 15. Land at 19. Check: 19 + 15 = 34.
Change in Quantity: Pictorial and Symbolic
Connect to the hundred chart: 47 + ? = 83. Start at 47 on the chart. Count by tens: 47, 57, 67, 77, 87. Too far by 4. So 30 + ? = 36: the ? is 36, but we need to subtract the 4 overshoot: 30 + 6 = 36. Total jump: 36. Check: 47 + 36 = 83.
Practice
Students solve 6 change problems (2 of each type), showing the visual model and the equation for each. Exit ticket: draw a ten-frame and write an equation to show 8 + ? = 15.
Exit ticket
Students solve 6 change problems (2 of each type), showing the visual model and the equation for each. Exit ticket: draw a ten-frame and write an equation to show 8 + ? = 15.
Step 1: Identify the structure before computing: start (26) → change (? flew away) → result (14). It's a change-unknown story.
Step 2: Draw the bar model instead of 26 little birds: one long bar labelled 26 on top; underneath, a bar of 14 (still here) and a mystery bar (flew away) that together match the 26.
Step 3: The drawing MAKES the equation: 14 + ? = 26 (or 26 − 14 = ?).
Step 4: Solve by climbing: 14 + 6 = 20, 20 + 6 = 26 → 12 birds flew away.
Step 5: Check inside the story: 26 birds, 12 leave, 14 remain ✓.
The pictorial stage is the bridge: students who can DRAW the structure of a story never have to guess add-or-subtract from keywords again.
Step 1: Notice nothing MOVES in this story — nobody gains or loses anything. It's not a change story; it's a COMPARISON. Many students add 35 + 19 because "more" appeared. The bars fix that.
Step 2: Draw two aligned bars: Maya's bar (35) on top, Jon's shorter bar (19) below, left edges lined up. The difference is the overhang — the part of Maya's bar that sticks out past Jon's.
Step 3: Compute the overhang: 19 + ? = 35. Climb: 19 + 1 = 20, 20 + 15 = 35 → 16.
Step 4: Answer in a full sentence: Maya has 16 MORE stickers than Jon. (And Jon has 16 fewer — same fact, other direction.)
The rule that matters: "how many more" is asking for the GAP between two amounts, never the total.
The story: 28 chairs are set up. The hall needs 45. How many more chairs?
Step 1: Let half the class write 28 + ? = 45 (the story as it unfolds: chairs being added) and half write 45 − 28 = ? (the calculation view: total minus what's there).
Step 2: Solve both. Adding up: 28 + 2 = 30, 30 + 15 = 45 → 17. Taking away: 45 − 28 = 17. Same answer — necessarily.
Step 3: Draw ONE bar model on the board: whole bar 45, parts 28 and ?. Point out that BOTH equations are readings of this single picture. The equations are two sentences describing one situation.
Step 4: The generalization to say out loud: every part-part-whole picture carries a family of equations, and you may solve whichever family member is friendliest. Choosing your equation is a solver's right.