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Math problem set Grade 4 Engineering & Design

Ashmill's New Filter Bed: Layer Depths in Tenths of a Foot

A grade 4 problem set at a fictional water plant where students add and subtract tenths of a foot and write filter layer depths as decimals.

Objectives

  1. Students will write a depth given in tenths of a foot using decimal notation.
  2. Students will add and subtract tenths with like denominators to find total layer depth and leftover space.
  3. Students will compare two filter designs by subtracting the depths of matching layers.

Vocabulary

Key words: filter bed, layer, tenth, depth, gravel

Teacher overview

The fictional Ashmill water plant is building a new filter box, and the crew has to stack three layers of gravel and sand inside it so that water trickles down through them while the dirt it carries stays behind in the sand. Crews mark depths on a measuring rod in tenths of a foot rather than in inches, so every layer in this set is given as a number of tenths, which students both add and rewrite as a decimal. All ten problems describe the same box, and problems 5 and 9 reuse the three-layer total from problem 4, which each of them restates so a student who erred earlier can still work the later items. A common error with this math is adding the denominators along with the numerators, writing 4/10 + 6/10 as 10/20; remind students that the tenths stay tenths because the layers are measured on the same rod.

Ashmill's New Filter Bed: Layer Depths in Tenths of a Foot — Grade 4 Engineering & Design
Name: ____________________________ Date: ______________

Problems

  1. The gravel layer in Design A is 4/10 foot deep. Write that depth as a decimal.
  2. The fine sand layer in Design A is 8/10 foot deep. Write that depth as a decimal.
  3. Design A has a gravel layer 4/10 foot deep and a coarse sand layer 6/10 foot deep. Add the two depths. Write the total as a fraction.
  4. Design A has three layers: gravel 4/10 foot, coarse sand 6/10 foot, and fine sand 8/10 foot. What is the total depth of all three layers? Write the total as a mixed number.
  5. The inside of the filter box is 2 feet deep, which is the same as 20/10 feet. The three Design A layers fill 1 8/10 feet. How much empty space is left above the layers?
  6. Design B has a gravel layer 3/10 foot deep, a coarse sand layer 6/10 foot deep, and a fine sand layer 9/10 foot deep. What is the total depth of Design B's three layers? Write the total as a mixed number.
  7. Design A has a fine sand layer 8/10 foot deep. Design B has a fine sand layer 9/10 foot deep. How much deeper is Design B's fine sand layer?
  8. In Design A, the fine sand layer is 8/10 foot deep and the gravel layer is 4/10 foot deep. How much thicker is the fine sand layer than the gravel layer?
  9. The crew adds one more layer of extra fine sand, 2/10 foot deep, on top of Design A's 1 8/10 feet of layers. What is the new total depth of the layers in the box?
  10. Rosa writes the fine sand layer as 8/10 foot. Dev writes the same layer as 0.8 foot. Explain why both students are right.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 0.4 foot. The fraction 4/10 means 4 tenths, and 4 tenths written as a decimal is 0.4.
  2. Answer: 0.8 foot. The fraction 8/10 means 8 tenths, and 8 tenths written as a decimal is 0.8.
  3. Answer: 10/10 foot. Add the numerators and keep the tenths: 4/10 + 6/10 = 10/10 foot, which is the same as 1 whole foot.
  4. Answer: 1 8/10 feet. First 4/10 + 6/10 = 10/10, then 10/10 + 8/10 = 18/10. Since 10/10 makes 1 whole foot, 18/10 feet is 1 8/10 feet.
  5. Answer: 2/10 foot. Subtract the layers from the box depth: 20/10 - 18/10 = 2/10 foot, which is 0.2 foot of empty space.
  6. Answer: 1 8/10 feet. First 3/10 + 6/10 = 9/10, then 9/10 + 9/10 = 18/10, and 18/10 feet is 1 8/10 feet, the same total as Design A.
  7. Answer: 1/10 foot. Subtract the two fine sand layers: 9/10 - 8/10 = 1/10 foot deeper in Design B.
  8. Answer: 4/10 foot. Subtract the gravel from the fine sand: 8/10 - 4/10 = 4/10 foot thicker.
  9. Answer: 2 feet. Add the new layer to the old total: 18/10 + 2/10 = 20/10 feet, and 20/10 feet is 2 whole feet, so the layers now fill the box with no space left above them.
  10. Answer: Both write the same depth in two ways. The fraction 8/10 means 8 tenths of a foot, and the decimal 0.8 also means 8 tenths of a foot, so the two numbers name one depth on the measuring rod.

Differentiation

Support

Give these students a number line from 0 to 2 marked in tenths and let them hop along it for the addition and subtraction items. Provide a strip of paper cut into ten equal parts to model one foot, and start with problems 1, 2, and 3 before the three-layer totals.

Extension

Ask these students to design a third filter bed that uses exactly 1 5/10 feet of layers in the same box, using three layers with depths written in tenths, then to state how much empty space their design leaves above the sand. Have them explain why more than one answer is possible.

ELL supports

Pre-teach "layer," "depth," and "tenth" with a labeled side-view drawing of the filter box showing gravel on the bottom and fine sand on top. Provide a sentence frame for problem 10: "The layer is ___ of a foot deep, and I can write that as ___ or as ___."

Extensions

Have students measure the depth of three stacked classroom objects, such as books, to the nearest tenth of a foot using a ruler marked in tenths, then record each depth as both a fraction and a decimal and add them the way the Ashmill crew adds filter layers.