Fill a table and complete a line graph from clues

5.MD.B.2 · adapt · grade 5

Archetype: Solve a Table or Graph Step by Step from Clues · step in a 5-type progression

Representative Problem

The line graph shows the number of picture frames a workshop made each month. In July the workshop made $80$ more frames than it made in June, and the number of frames made in July and August together is $640$ . Complete the line graph.

(Figure) A line graph titled "Picture Frames Made." The horizontal axis shows the month the count was taken — $4$ (April), $5$ (May), $6$ (June), $7$ (July), and $8$ (August). The vertical axis shows the number of frames, with major gridlines at $0$ , $100$ , $200$ , $300$ , and $400$ and each small grid square worth $20$ frames. Points are plotted and joined for April at $280$ , May at $380$ , and June at $180$ ; the points for July and August have not yet been plotted.

Show solution

Understand

A line graph of picture frames made each month shows April = 280, May = 380, June = 180; July and August are blank. July is 80 more than June, and July + August together is 640. I must find the July and August counts to complete the graph.

Givens

April = 280 frames, May = 380 frames, June = 180 frames
July = June + 80
July + August = 640
Each small grid square = 20 frames

Unknowns

Number of frames made in July
Number of frames made in August

Constraints

Plotted July and August values must land on grid squares worth 20 each

Plan

#7 Identify Subproblems · also uses: #11 Work Backwards

I solve in order from what I can already know: first July (June plus 80), then August (the 640 total minus July). Each is a one-step subproblem, and finding August uses the total backwards.

Execute

#7 Identify Subproblems 4.OA.A.3

July is 80 more than June, and June is 180, so July = 180 + 80 = 260.

180 + 80 = 260

Start from the value you already know (June) and apply the given step.

#11 Work Backwards 4.OA.A.3

July and August together make 640, so August = 640 - July = 640 - 260 = 380.

640 - 260 = 380

Knowing the total and one part, subtract to recover the other part.

#7 Identify Subproblems 5.MD.B.2

On the graph (each square = 20), July 260 sits 3 squares above 200, and August 380 sits 4 squares above 300; join June-July and July-August with straight segments.

260 = 200 + 3\times 20,\quad 380 = 300 + 4\times 20

Converting the numbers into squares lets you place each point exactly.

Answer: July = 260 frames, August = 380 frames

Review

Check the clues: July 260 is indeed 80 more than June 180, and July 260 + August 380 = 640 as required. Both values land neatly on grid lines (260 and 380 are multiples of 20), so the plotted points are valid.

Convert to algebra (tool 13): let July = J = 180 + 80 and August = 640 - J; solving gives the same J = 260 and August = 380.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Computing July from June and August from the total
5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Placing the July and August values on the scaled graph

💡 Fill in what you can figure out first, then use the total to get the rest - then just plot!